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Photpgraphic 

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errata 
to 


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1 

2 

3 

4 

5 

6 

-Jk  <. 


-^'ISfv'iv'fi''^* 


Sr  TBS  SAMJE  AUTHOft. 


AN  ELEMENTARY  TREATISE  ON  ANALYTIC  GEOMETRY, 

IMBKACINa 

'■       -         PLANE    GEOMETRY, 

AMD  AN 

INTRODUCTION    TO   QEOMETRV    OF  THREE   OlMENtlONS. 


AN    ELEMENTARY    TREATISE 


DIFFERENTIAL    AND     INTEGRAL    CALCULUS. 


WITH    NUMEROUS    EXAMPLES. 


/ 

J1 


«te 


cGEOMETRv,  I  ANALYTIC  MECHANICS. 


IIMENtlONS. 


ISE 


:alculus. 


WITH    NUMEROUS    EXAMPLES. 


EDWARD  A.  BOWSER,  LL.D., 


riovMKa  or  mathematics 


KUTCnS    COLLKW. 


NEW    YORK: 

D.     VAN      NOSTRAND, 
tB  McnuT  AKD  ft  WAina  Sranr, 

1884. 


5    "f 


..■*  4:  X.-     ■ 


T' 


OOPTSXOBT,  t88k,  BT  E.  A.  BOWSXS. 


V  ■  Jj"v?j''"'/'-:.^^/,*;y: 


c^ 


^te 


PREFACE. 


"0*^ 


■♦»»■■ 


rpHE  preoent  work  on  Analytic  Mechanics  or  Dynamics  is  desired 
as  a  text-book  for  the  mudents  of  Scientific  Schools  and  Col. 
leges,  who  have  received  training  in  the  elements  of  Analytic  Oeome- 
try  and  the  Calculus. 

Dynamics  is  here  used  in  its  tme  sense  as  the  science  of  fortt. 
Tlie  tendency  among  the  best  and  most  logical  writers  of  the  present 
(lay  appears  to  l)e  to  use  this  torm  for  the  science  of  Analytic  Me- 
chanics, while  the  branch  formerly  called  Dynamics  is  now  termed 
Kinetics. 

The  treatise  is  intended  especially  for  beginners  in  this  branch  of 
Hcieuce.  It  involves  the  use  of  Analytic  Geometry  and  the  Calculus. 
The  analytic  method  has  been  chiefly  adhered  to,  as  being  better 
adapted  to  the  treatment  of  the  subject,  more  general  in  its  applica- 
tion and  more  fruitful  in  results  than  the  geometric  method ;  and  yet 
wliere  a  geometric  proof  seemed  preferable  it  has  been  introduced. 

The  aim  has  been  to  make  every  principle  clear  and  intelligible, 
to  develop  the  di£Ferent  theories  with  simplicity,  and  to  explain  fully 
the  meaning  and  use  of  the  various  analytic  expressions  in  which  the 
|)rinciple8  are  embodied. 

The  book  ocmsists  of  three  parts.  Part  I,  with  the  exception  of  a 
preliminary  chapter  devoted  to  definitions  and  fundamental  princi* 
pies,  is  entirely  given  to  Static*. 

Part  II  is  occupied  with  Kinematics  and  the  principles  of  this 
important  branch  <>f  mathematics  are  so  treated  that  the  student  may 
enter  upon  the  study  of  Kinetics  with  clear  notions  of  motion,  veloc- 
ity and  acceleration.  Part  III  treats  of  the  KineUcs  cf  a  particle  and 
of  rigid  Imdiee. 


tW" 


!^^!^'S^-f_ '  t.'; Jnu;  1  ^."Vg!" 


'  W^:    :;;^:;,;^i----^.    -^  PREFACE.    - 

In  this  arrangement  of  ti.««  work,  with  the  exception  of  Kine- 
matics, I  have  followed  the  plan  naually  adopted,  and  made  the 
subject  of  Statics  precede  that  of  Kinetics. 

For  the  attainment  of  that  grasp  aX  principles  which  it  is  the 
special  aim  of  the  book  to  impart,  numerous  examples  are  given  at 
the  ends  of  the  chapters.  The  jyreater  part  of  thorn  will  present  no 
serious  difficulty  to  the  student,  while  a  few  may  tax  his  best 
efforts. 

In  prepwring  this  book  I  have  availed  myself  of  the  writings  of 
many  of  the  beat  authont  The  chief  sources  frtm  which  I  have 
derived  assistance  are  the  treatises  of  Price,  Miuchin,  Todhunter, 
Pratt.  Routh,  Thomson  and  Tait,  Tait  and  Steele,  Weisbach,  Venta- 
roli,  Wilson,  Browne,  Gregory,  R»nkine,  BoucharUt,  Pirie,  Lagrange, 
and  La  Place,  while  many  valuable  hints  as  well  as  examples  have  been 
obtained  from  the  works  of  Smith,  Wood,  Bartlett,  Young,  Moeek/, 
Tate,  Magnus,  tioodeve,  Parkinson,  Olmsted,  Gamett,  Benwick,  Bot- 
tomley,  Morin,  Twisden,  Whewell,  Oalbraith,  Ball,  Dana,  Byrne,  ihe 
Elusjyclopedia  Britannica,  and  the  Mathematical  Visitor. 

1  have  again  to  thank  my  old  pupil,  Mr.  B.  W.  Pi«ntiss,  of  the 
Nautical  Almanac  Office,  and  formeriy  Fellow  in  Mathematics  at  the 
Johns  Hopkins  University,  fbr  reading  the  MS.  and  for  valuable  sug. 
gestiona  Several  others  also  of  my  friisnds  have  kindly  assisted  me 
by  correcting  proofi^beets  and  verifying  copy  aad  foimule. 

E.  A.  B. 
BtrroBRS  Collbob,  i 

Nbw  Bbdhbwick,  N.  J.,  June,  1884. ' 


eptlon  of  Kine- 
and  made  tho 


which  it  is  tlio 
ilea  are  given  at 
will  present  no 
J  tax  his  best 


TABLE  OF  CONTENTS. 


'  the  writings  of 
I  which  I  have 
lin,  Todhunter, 
eisbacb,  Yentu- 
'irie,  Lagrange, 
nples  have  been 
oung,  Moselc^ , 

Benwick,  Bot. 
ma,  Bjrrne,  the 
r. 

'rentiss,  of  the 
ematios  at  the 

valuable  aug. 
aasisted  me 
vim. 
E.  A.  B. 


PART   I. 


lly 


CHAPTER    I 

FIRST    PBINCIPLE8. 

m.  '^» 

1.  Definitions— Statics,  Kinetics  and  Kinematies 1 

2.  Matter * 

8.  InerJa * 

4.  Body,  Space  and  Time 8 

6.  Rest  and  Motion 8 

6.  Velocity •• 8 

7.  Pormube  for  Velocity * 

b.  Acceleration ■ * 

9.  Measure  of  Acceleration S 

10.  Geometric  Representation  of  Velocitj  and  Accelerat'on 6 

11. -Mass * 

12.  Momentum "^ 

18.  Change  of  Momentum ' 

14.  Force ® 

15.  Static  Measure  of  Force 8 

16.  Action  and  Reaction ® 

17.  Method  of  Comparing  Forces • 

18.  Representation  of  Forces 1® 

19.  Measure  of  Accelerating  Forces 10 

20.  Kinetic  Measure  of  Force H 

31.  Absolute  or  Kinetic  Unit  of  Force 18 

22.  Throe  Ways  of  Measuring  Force 14 

28.  Meaning  of  fli  In  Dynamics l** 


IS'., 


rr.,,  ".i'W.^g!^. 


'#■ 


roirrxyrs. 


24.  OniTltation  Units  Df  Force  and  Mam...... ic 

25.  Gravitation  and  Absolute  Measure 17 

Examples 18 


■    STATICS     (REST).     -^ 


CHAPTER    II. 


THE      COMPOSITION      AND      BE80LUTI0N      OF     CONCUBBINO 
FOROKS — CONDITIONS    OF    EQUILIBBIUM. 

26.  Problem  of  Statics 21 

27.  Concurring  and  Conspiring  Forces 21 

28.  Composition  oi  Conspiiring  Forces 22 

29.  Composition  of  Velocities 23 

30.  Composition  of  Forces 24 

81.  Triangle  of  Forces. 25 

82.  Three  Concurring  Forces  in  Equilibrium 26 

83.  The  Polygon  of  Forces 27 

84  Paralleloplped  of  Forces 28 

85.  Resolution  of  Forces 80 

86.  Magnitude  and  Direction  of  Resultant 81 

87.  Conditions  of  Equilibrium 32 

88.  Resultant  of  Concurring  Forces  in  Space 84 

89.  Equilibrium  of  Concurring  Forces  in  Space 85 

40.  Tenaion  of  a  String 85 

41.  Equilibrium  of  Concurring  Forces  on  a  Smooth  Plane 39 

42.  Equilibrium  of  Concurring  Forces  on  a  Smooth  Surface 41 

Examples 45 


CHAPTER    III. 


COMPOSITION    AND    BKSOLCTION    OF  FOBCES    ACTING   ON  A 

RIGID  BODY. 

48.  ARlgldBody 57 

44.  Transmissibility  of  Force 57 

46.  Resultant  of  Two  Parallel  Forces 58 

46.  Moment  of  a  Force flo 


'.*!a3i-Bia»»»»u.im.'. 


-.v\aiifevj^v.a».£?i<ii<»aB«iiii<fe';-fa'n>^>&<fe^ 


^te 


PAO« 

It 

17 

18 


CONCUBBIlfO 
IIUJI. 

21 

21 

22 

28 

24 

25 

26 

27 

28 

80 

81 

32 

84 

85 

86 

lane 30 

urface 41 

46 


iCTINQ  ON  A. 


57 

57 

60 


COm'JBJfTS. 


»a 


AST.  VASI 

47.  Signs  of  Moments. , 61 

48.  Geometric  Repreeent&tion  of  a  Moment 61 

40.  Two  £qual  anU  Opposite  Panllel  Foroea 61 

5U.  Moment  of  a  Couple 62 

51.  Effect  of  B  Couple  on  a  Rigid  Body 68 

52.  Efiect  of  Transferring  Couple  to  Parallel  Plane  not  altered.. .  64 
68.  A  Couple  replaced  by  another  Couple 61 

54.  A  Force  and  a  Couple 06 

55.  Resultant  of  any  number  of  Couples 66 

56.  Resultant  of  Two  Couples. 07 

57.  Varignon'H  Thiorem  of  Moments 69 

58.  Varignon'a  Theorem  for  Parallel  Foices 71 

59.  Centre  of  Parallel  Forces 71 

6C.  Equilibrium  of  a  Rigid  Body  under  Pcrallel  Forces. 74 

61.  Equilibrium  of  a  Rigid  Body  under  Fozoes  in  any  Direction. .  76 

62.  Equilibrium  under  Tliree  Forces. 77 

63.  Centre  of  Parallel  Forces  in  hifferent  Planes 86 

64.  Equilibrium  of  Parallel  Forces  in  Space. 86 

65.  Equilibrium  of  Forces  acting  in  any  Direction  in  Space 88 

Examples 00 


CHAPTER    IV. 

OBNTRB  OF  GRAVITY   (CENTRE  OP  KASS). 

66.  Centre  of  Gravity 100 

67.  Planes  of  Byiametry — Axes  of  Symmetry 101 

68.  Body  Suspended  from  a  Point ICl 

00.  Body  Supported  on  a  Surface 102 

70.  Different  Kinds  of  Equilibrium 102 

71.  Centre  of  Gravity  of  Two  Masses 108 

72.  Centre  of  Gravity  of  Part  of  a  Body 108 

78.  Centreof  Gravity  of  a  Triangle 108 

74.  Centre  of  Gravity  of  a  Triangul.ir  Pyramid 106 

75.  Centre  of  Gravity  of  a  Cone 106 

76.  Centre  of  Gravity  of  Frustum  of  Pyramid 107 

77.  Investigations  involving  Integration 100 

78.  Centre  of  Gravity  of  the  Ate  of  a  Curve 110 

79.  Centre  of  Gravity  of  a  Plane  Ai8<i 116 

80.  Polar  Elements  of  a  Plane  Area 118 


•  •• 

VUl 


CONTENTS. 


Ater. 
81. 
82. 
88. 

84. 
85. 
8«. 
87. 


go. 


PAO» 

Double  Integration— Polar  FomiQlae 120 

Doiibln  Integration — Rectangular  Pormulte 122 

Centre  of  Gravity  of  a  Surface  of  Kevolution. 123 

Centre  of  Gravity  of  any  Curved  Surface 126 

Centre  of  (Jruvity  of  a  Solid  of  Revolution 127 

Polar  Fonnu'ffl • 130 

Centre  of  Gravity  of  any  I'felid ]3i 

Polar  Elements  of  Mass 188 

Special  Methods isg 

Theon-ms  of  Pappus igg 

Examplui 140 


CHAPTER    V.  • 

,  -  '■•■   :*':■'  FRICTIOK.  ;.'  '"•>":'''■     " 

91.  Friction 149 

92.  Laws  of  Friction 150 

93.  Magnitudes  of  Coefflcionts  of  Friction i(  i^ 

94.  Angle  of  Friction ^^.^ 

95.  Reaction  of  a  Rougli  Curve  or  Surface 163 

96.  Friction  on  an  Inclined  Plane 154 

97.  I'riction  or.  a  Doable  Inclined  Plane 156 

98.  Friction  on  Two  Inclined  Planes. 159 

99.  Friction  of  a  Trunnion 159 

100.  Friction  of  a  Pivot .,,,  y-Q 

Examples 192 


CHAPTER    VI. 


THE  PRINCIPLE  OF  VIRTUAL   VELOCITIE& 

101.  Virtual  Velocity igg 

102.  Principle  of  Virtual  Velocities 167 

103.  Nature  of  the  Displacement 169 

104.  Equation  of  Virtual  Momenta '. , 169 

105.  System  of  Particles  Rigidly  Connected 170 

Examples 172 


120 
122 
123 
126 
127 
130 
181 
183 
136 
188 
140 


149 

J50 

1/-: 

ib'a 
153 
154 
156 
159 
159 
ICO 
162 


coifTEyrs. 


CHAPTER    VII. 

.  "■■   „  MACHINES.     „  %' 

u».    -     ■■  -    '  -"'■•»" 

106.  Functions  of  a  Machine 177 

107.  Mechanical  /  Jvantage 178 

108.  Simple  Machinee , • 1^0 

109.  Th<i  Lever '. !..... 181 

110.  Equilibrium  of  the  Lever 181 

111.  The  Common  Balance 184 

112.  Chief  Reqaisites  of  a  Good  Balance 186 

113.  The  Steelyard 188 

114.  To  Graduate  the  Common  Steelyard 188 

116.  The  Wheel  and  Axle 190 

116.  Equilibrium  of  the  Wheel  and  Axle 190 

117.  Toothed  Wheels 192 

118.  Belation  of  Power  and  Weight  in  Toothed  Wheels 198 

119.  Rtlation  of  Power  to  Weight  in  a  Train  of  Wheels. 194 

120.  Thfl  Inclined  Plane IWJ 

121.  The  Pulley 1»7 

122.  The  Simple  Movable  Pulley 198 

123.  Firsi  System  of  Pulleys. • 1*8 

124.  Second  System  of  Pulleys 200 

126.  Third  System  of  Pulleys 201 

126.  The  Wedge. 20S 

127.  Mechanical  Advantage  of  the  Wedge 202 

128.  The  Screw 304 

129.  Relation  between  Power  tod'  Weight  in  the  Screw 204 

129a.  Prony's  Differential  Screw 206 

Examples 207 


166 
107 
169 
169 
170 
172 


CHAPTER    VIII. 

THE   FtTNICULAR  POLYGON — THE  CATENARY — ATTRACTION. 

190.  Equilibrium  of  the  Funicular  Polygon 216 

131.  To  Construct  the  Funicular  Polygon 218 

182.  Cord  Supporting  a  Load  Uniformly  Distributed 219 

188.  The  Common  Catenary — Its  Equation 221 

188a.  Attraction  of  a  Spherical  Shell 226 

Examples 228 


1 1  ^, ,■■«  mfiijn|i,»ifyj^vi^j;i^^iLWLii_^wi ir,!  ■-- ->,  i*ii«ii;i.'i.y.L.ji  •n-it'yjr!>f'fti;Ti^i''viT-r'yKfr'!!mi'^.,J  V i'-!tf 'g^'!'^r>l!?*P^'7'iy 


coAnuvTS. 


PART   II. 
KINEMATICS    (N.OTION). 


CHAPTER    I. 


RECTILINEAB     MOTION. 
ABT.  r^ag 

184.  DefinltlonB— Velocity 281 

185.  Acceleration 288 

188.  Relation  lKtwe«n  Space  and  Time  when  Aooeleration  =  0.. .  288 

187.  Relation  when  the  Acceleration  is  Constant 284 

188.  Relation  when  Acceleration  Tariea  as  the  Time 235 

189.  Relation  waen  Acnolenttion  Vfrico  ai>  the  Distance 285 

140.  Ekiuations  of  Motion  for  Falling  Bodies. 287 

141.  Particle  Piv^jected  Vertically  Upwards 280 

142.  Oompoeitions  of  Velodtias 242 

148.  Resolution  of  Velodtiea. 248 

144  Motion  on  an  Inclined  Plane 245 

146.  Times  of  Descent  down  <'^ords  of  a.  Circle 247 

146.  The  atraight  Line  of  Quickest  Descent 248 

Examples. , 249 


CHAPTER    II 

CCEVItlNEAE    MOTHyN. 

147.  Remarks  on  Curvilinear  Motion 258 

J48.  Composition  of  Unifo.m  Velocity  and  Acceleration 258 

140.  C3m  position  and  Resolution  of  Acceleration 259 

Examples 2fli 

160.  Motion  «♦  Project llos  '•»  Vacuo 266 

161.  The  Path  of  a  Particle  in  Vacuo  <s  k.  Parabola 266 

162.  The  Parameter— Range— Greatest  Height— Height  oi  Direc 

trix 267 

188.  Velocity  of  a  Particle  at  an.-  point  of  its  Path 260 

164.  Time  of  Flight  along  a  Horisoiitn!  Plane 260 

166.  Poiut  at  which  a  Projectile  will  Strike  ac  IncUnad  PImm.  . .  270 


Klh 


LAWS 
VA 

166.  D 
166   N 

167.  R 
168. 'K 

169.  R 

170.  T 

171.  W 

172.  M 
178.  M 

174.  V 

175.  M 
170.  \ 

177.  V 

178.  \. 
170.  M 

E 


CONTSKTa. 


XI 


o. 


PAOK 

Projection  for  Grenteat  Butg*  on  a  Given  Plane 370 

The  Elevation  that  the  Particle  maj  pen  n  Given  Point. . .     271 

Second  Method  of  Finding  E^.nation  of  Trajectory 273 

Velocity  of  Discharge  of  Bails  and  Shells 274 

Angular  Vilocity  and  Angular  Acceleration 275 

Accelerations  Along  and  Perpendicular  to  RadiuB  Vector. . . .  278 

Acceleraiions  Along  and  Perp<^ndicular  to  Tangent 279 

When  Acceleration  Perpendicular  to  Radios  Vector  is  zero  .  281 

When  Angular  Velocity  is  Constant 282 

Examples 284 


PART    III. 
KINETICS    (MOTION    AND    FORCE). 


858 

258 

259 

....     261 

206 

866 

u  Direc- 
....  267 
sin 

....  869 
lane...  870 


CHAPTER    I. 

LAWS    OP    MOTION — MOTION     UNDEB     XHF     ACTION     OF     A 
VARIABLE  FORCE- -MOTION   IN   A    RESISTING   MEDIUM. 

165.  Definitions 289 

166   Newton's  Laws  of  Motion 289 

167.  Remarks  on  taw  I , 890 

168.  "Remarks  on  Law  II 291 

169.  Komnrks  on  Law  III 894 

170.  Two  LawD  of  Motion  in  the  French  Treatises 895 

171.  Motion  of  Particle  under  an  Attractive  Force 295 

172.  Motion  under  the  action  of  a  Variable  Ropnlsive  Force 298 

178.  Motion  under  the  action  of  an  Attractive  Force ^1)9 

174.  Velocity  acquired  In  Falling  through  a  Great  Height 800 

176.  Motion  in  a  Resisting  Medium.  ;!02 

176.  Motion  in  the  Air  against  the  Action  of  Gravity 804 

177.  M^ion  of  a  Projectile  in  a  Resisting  Medium 807 

178.  Motion  against  the  Resistance  of  the  Afnuitiphere 908 

179.  Motion  in  the  Atmoephere  under  a  smuii  Angle  of  Elevation  812 
Examples 818 


zU 


CONTENTS. 


CHAPTER    II. 

CENTRAL  FORCES. 

180.  I>?flnition8 321 

181.  A  Particle  under  the  Action  of  a  Central  Attraction 821 

182.  The  Sectorial  .\rea  Swept  over  by  the  Radius  Vector 825 

183.  Velocity  of  Particle  at  any  Point  of  it«  Oruit 826 

184.  Orbit  when  Attraction  as  the  Inverse  Square  of  Distance. . .  32 J 
186.  Suppoee  the  Orbit  to  be  an  Ellipee 833 

186.  Kepler's  L»w8 335 

187.  Nature  of  the  Force  which  acta  upon  the  Planetary  System.  885 
Kxampies ....  338 


CHAPTER    III. 

CONSTRAIKED    MOTIOK 

188.  Deflnitions 848 

189.  Kinetic  Energy  or  Vis  Viva— Work 345 

190.  To  Find  the  Heaetion  of  the  Constraining  Curve 848 

101 .  Point  where  Particlj  will  leave  Constri  ining  Curve 849 

192.  Constrained  Motion  Under  Action  of  Oraviiy 350 

108.  Motion  on  a  Circular  Arc  in  a  Vertical  Plane 850 

194.  The  Simple  Pendulum 353 

196.  Relation  of  Time,  Length,  and  Force  of  OravHy 858 

196.  Height  of  Mountain  Determined  with  Pendulum ;t64 

197.  Depth  of  Mine  Determined  with  Pendulum 866 

198.  Centripetal  and  Centrifugal  Forces 866 

199.  The  Centrifugal  Force  at  the  Equator 058 

200.  Centrifugal  Force  at  Diffcn>nt  Latitudes 869 

901.  The  Conical  Pendiilnm— The  Uovemor 861 

Examples 992 


CHAPTER    IV, 

IMPACT. 

209.  An  Impulsive  Porce 370 

208.  Impact  i)r  Collision 371 

204.  Direct  and  Central  Im|4u;t 872 

206.  Elaaticity  of  Bodies— Coefficient  of  Restitution 878 


ABT. 

20(5. 
','07 
208. 
2()9. 
210. 


211. 
212 
213. 

214. 
215. 

210.  ] 
] 

217.  1 
1 

218.  I 

219.  i 

230.  \ 

231.  1 
222.  > 
238.  6 

I 


224  1 

226.  8 

236. 

237. 

238. 

239. 


230.  I 

231.  J 


iilif  I 


^ 


CONTSNTa, 


xm 


...   821 

Ion S21 

ector 825 

826 

Distance. . .  32 J 

888 

385 

aiy  System .  885 


848 

845 

848 

irvo 849 

830 

850 

, 858 

858 

, :i54 

355 

856 

aw 

859 

861 


870 
871 
872 
878 


ABT.  rAOB 

30«.  Direct  Impact  of  Inelastic  Bodies 374 

'207.  Direct  impact  of  Eliifltic  Bodien 875 

-m.  Loss  of  Kinetic  Energy  in  Impact  of  Bodies 878 

2()9.  Oblique  Im|jact  of  Bodies 880 

210.  Oblique  Impact  of  Two  Smooth  Spheres 883 

Examples 888 


CHAPTER    V. 

WOEK  AND  ENERGY. 

Definition  and  Meaanre  of  Work 880 

General  Case  of  Worli  done  by  a  Force l>90 

Work  on  an  Inclined  Plane 891 

Examples 808 

Horse  Power 895 

Work  of  Raising  a  System  of  Weights ?0« 

Examples 8i»7 

Modulus  of  a  Machine 400 

Examples 401 

Kinetic  and  Potential  Energy— Stored  Work 404 

Examples 406 

Kinetic  Energy  of  a  Rigid  Body  Revolving  roond  an  Axis. . .  403 

Force  of  a  Blow 411 

Work  of  a  Water  Fall 418 

The  Duty  of  an  Engine 414 

Wr.k  of  a  Variable  Force 415 

Sitiipeon's  Rule 415 

Exampltw 417 


211. 
212 
213. 

214. 
215. 

210. 

217. 

218. 
219. 
230. 
231. 
223. 
233. 


234. 
225. 
226. 
227. 
238. 
229. 

230. 
231. 


CHAPTEK    VI. 

MOMENT  UF   INERTIA. 

Moment  of  In»«rtia 480 

Moments  of  Inertia  relative  to  Paral'pl  Axes  or  Planes 438 

Radius  of  Gyration 434 

Polar  Moment  of  Inertia 4ii6 

Moment  of  Inertia  of  a  Solid  of  Revolution 487 

Moment  of  Inertia  aliout  A:  is  Perpendicular  to  Geometric 

Axis 438 

Moment  of  Inertia  of  Various  Solid  Bodies 440 

Moment  of  Inertia  of  a  Lamina  with  respect  to  any  Axis. . . .  441 


'X-f.  ;jy-"'j;i'v4^;f?g'^svj?!gy 'l^w'^:^M^l■fti.^l  jija^f[^^4»^      ,i,\UAm,v:'^>!WSSS^^ 


1^ 


lir 


CONTMNTB. 


232.  Principal  Axes  of  E Body.... 448 

2S3.  Products  of  Inortia ■■ 446 

Examples 447 


^ 


Hi 


CHAPTER    VII. 

ROTATORY   MOTION. 

284.  Imprefwed  and  Effective  Forces 461 

285.  D'Alembert'B  Principle 453 

286.  RoUtion  of  a  Rigid  Body  about  a  Fixed  Axis 454 

287.  The  Compound  Pendulum 457 

28M.  Length  of  Second's  Pendulum  Determined  Experimentally. .  462 
289.  Motion  of  a  Body  when  Unconstrained 4t?4 

240.  Centre  of  Percussion— Axis  of  Spontaneous  Rotation 404 

241.  Principal  Radius  of  Gyration  Determined  Practically 467 

242.  The  Ballistic  Pendulum 468 

248.  Motion  of  a  Body  about  a  Horizontel  Axle  through  its  Centre  470 

244.  Motion  of  a  Wheel  and  Axle 471 

245.  Motion  of  a  Rigid  Body  about  a  Vertical  Axis 472 

246.  Body  Rolling  down  an  ..uclined  Plane 478 

247.  Falling  Body  under  an  Impulse  not  through  its  Centre 475 

Examples 477 

CHAPTER    VIII. 

MOTION  OF  A  SYSTEM   OF   RIGID  BODIES  IN  BPAOB. 

248.  Equations  of  Motion  obtfline<l  by  D'  '.Icmbert's  Principle 481 

249.  Independence  of  the  Motions  of  Translation  and  RoUtiou. . .  482 

250.  Principle  of  the  Conservation  of  the  Centre  o£  Gravity 486 

251.  Principle  of  the  Conservation  of  Areu 486 

253.  Conservation  of  Vis  Viva  or  Energy 488 

253.  Composition  of  Rotations. 493 

254.  Motion  of  a  Rigid  Body  referred  to  Fixed  Axes 494 

866.  Axis  of  Instantanaoos  Rotation 495 

266.  Angular  Velocity  about  Axis  of  Instantaneous  Rotation 496 

257.  Eulor's  liquations ^^ 

258.  Motion  alx)Ut  a  Princi|»al  Axis  through  Centre  of  Gravity. . .  499 
269.  Vel<,.lty  abouta  Principal  Axis  when  Accelerating  Forces 

=  0 Ml 

860.  The  Integral  of  Euler's  Equations 502 

Examples "** 


, 44S 

446 

447 

451 

462 

454 

457 

Imentallv,.  462 

4«i4 

an 404 

illy 467 

468 

liita  Centre  470 

471 

472 

478 

^ntre 475 

477 


N  SPACE. 

indple...  461 
lotation...  482 

ivity 485 

486 

488 

498 

494 

405 

iation 400 

407 

Iravity...  499 
ng  Forces 

601 

603 

505 


ANALYTIC    MECHANICS. 


PART    I. 


CHAPTER    I. 

FIRST    PRINCIPLES. 

1.  Deflzdtioiis. — Analytic  Mechanics  or  Dynamics  ia 
the  science  which  treats  of  the  equilibrium  an<J  motion 
of  bodies  under  the  action  of  force.  It  is  accordingly 
divided  into  two  parts,  Statics  and  Kinetics. 

Statics  treats  of  the  equilibrium  of  bodies,  and  the  condi- 
tions governing  the  forces  which  produce  it 

Kinetics  treats  of  the  motion  of  bodies,  and  the  laws  of 
the  forces  which  produce  it. 

The  consideration  that  the  properties  of  motion,  velocity, 
and  displacement  may  be  treated  apart  from  the  particular 
forces  producing  them  and  independently  of  the  bodies  sub- 
ject to  them,  has  given  rise  to  an  auxiliary  branch  of  Dyna- 
mics called  Kinematics.* 

Although  Kinematics  may  not  be  regarded  as  properly 
in'^luded  under  Dynamics,  yet  this  brunch  of  science  is  so 
important  and  useful,  and  its  application  to  Dynamics  so 
immediate,  that  u  jwrtion  of  this  work  is  devoted  to  its 
treatment. 

*  Thli  name  wm  given  by  Amp4re. 


1  MATTES,  INERTIA,  BODY,  MOTION,  ETC. 

Kinematics  is  the  science  of  pore  motion,  withont  refer- 
ence to  matter  or  force.  It  treats  of  the  properties  of 
motion  without  regard  to  what  is  moving  or  how  it  ia 
moved.  It  is  an  extension  of  pure  geometry  by  introduc- 
ing the  idea  of  time,  and  the  consequent  idea  of  velocity. 

2.  Matter. — Matter  is  that  which  can  be  perceived  by 
the  senses,  aud  which  cia  transmit,  and  be  acted  upon  by 
force.    It  has  extension,  resistanon,  and  impenetrability. 

A  definition  of  matter  which  wonld  ntiefy  the  metaphyR'.cian  Is  ■ 
not  required  for  this  work.    It  ia  Bofflcient  for  na  to  oouceive  of  it  aa 
capable  of  receiving  and  transmitting  force :  becanoe  it  la  in  thia 
aspect  onlj  that  it  ia  of  importance  in  the  preaent  treatise. 

3.  Inertia. — By  Inertia  is  meant  that  property  of  mat- 
ter by  which  it  remains  in  its  state  of  rest  or  uniform 
motion  in  a  right  line  unless  acted  apon  by  force.  Inertia 
expresses  the  iact  that  a  body  cannot  of  itself  change  its 
condition  of  rest  or  motion.  It  follows  that  if  a  body 
change  its  state  from  rest  to  motion  or  from  motion  to  rest, 
or  if  it  change  its  direction  from  the  natural  rectilinear 
path,  it  must  have  been  influenced  by  some  external  cause. 

4.  Body,  Space,  and  Time.— ^  Body  is  a  portion  of 
matter  limited  in  every  direction,  and  is  consequently  of  a 
determinate  form  and  volume. 

A  Rigid  Body  is  one  in  whlvh  the  relative  positions  of 
its  particles  remain  unchanged  by  the  action  of  forces. 

A  Particle  is  a  body  indefinitely  small  in  every  direction, 
and  though  retaining  its  material  propertias  may  be  treated 
as  a  geometric  point 

Space  is  indefinite  extension.  Time  is  any  limited  por- 
tion of  duration. 

5.  Rest  and  Motion.— A  body  is  at  rest  when  it  con- 
stantly occupies  the  same  place  in  space.    A  body  is  in 


i 


TO. 


VMLOCtTT. 


ithont  refer- 
roperties  of 
>r  how  it  is 
by  introduo- 
f  velocity. 

aerceived  by 
;ed  upon  by 
etrability. 

ttaphyo'.cian  Is  . 
uceive  at  it  u 
I  it  is  in  this 
iae. 

jerty  of  mat- 
;  or  uniform 
rce.  Inertia 
If  change  its 
it  if  a  body 
otion  to  rest, 
il  rectilinear 
temal  cause. 

a  portion  of 
uently  of  a 

positions  of 
forces. 

|ry  direction, 
Ly  be  treated 

limited  por- 


rhen  it  con- 
body  is  in 


motion  when  the  body  or  its  parts  occupy  successively  dif- 
furent  positions  in  space.  But  we  cannot  judge  of  the  stute 
of  rest  or  motion  of  a  body  without  referring  it  to  the 
]K)iiition8  of  other  bodies  ;  and  hence  rest  and  motion  must 
be  considered  as  necessarily  relative. 

If  there  were  anything  which  we  knew  to  be  absolately  fixed  in 
space,  we  mi^t  perceira  aiieolate  motion  by  change  of  place  with 
rcfertince  to  that  object.  Bat  as  we  know  of  no  such  thing  as  also- 
lute  rest,  it  follows  that  all  motion,  as  measured  1>t  ub,  must  be 
rolatiye ;%.».,  most  relate  to  something  which  we  assume  to  be  fixed. 
Hence  the  same  thing  may  often  I>e  itaid  to  be  at  rest  and  in  motion 
at  the  same  time :  for  it  may  be  at  rest  in  regard  to  one  thing,  anJ.  in 
niution  in  regard  to  another.  For  example,  the  objects  on  a  vessel 
may  be  at  rest  with  reference  to  each  other  and  to  the  voseel,  while  they 
are  in  motion  with  reference  to  the  neighboring  shore.  So  a  man, 
I>unting  his  barge  up  the  river,  by  leaning  against  a  pole  which  rests 
on  the  bottom,  and  walking  on  the  deck,  is  In  motion  relative  to  the 
barge,  and  in  motion,  but  in  a  different  manner,  relative  to  the  car- 
rui}t,  while  he  is  at  rest  relative  to  the  earth. 

Motion  is  uniform  when  the  body  passes  over  equal  spaces 
iu  equal  times ;  otherwise  it  is  variable. 

6.  Velocity. — The  velocity  of  a  body  is  its  rate  of 
motion.  When  the  velocity  ic  eonstant,  it  is  measured  by 
the  space  pas^-ed  over  in  a  unit  of  time.  When  it  is  varia- 
ble,  it  is  measured,  at  any  instant,  by  the  space  over  which 
the*  body  would  pass  in  a  unit  of  time,  were  it  to  move, 
during  that  unit,  with  the  same  velocity  that  it  has  at  the 
instant  considered. 

The  speed  of  a  railway  train  is,  in  general,  variable.  If  we  were  to 
Bay,  for  example,  that  it  was  running  at  the  rate  of  80  miles  an  hour, 
W6  would  not  mean  that  it  ran  80  miles  daring  the  last  hour,  nor  that 
it  would  run  80  miles  daring  the  next  hour.  We  would  mean  that,  if 
it  were  to  ruu  for  an  hour  with  the  speed  which  it  now  has,  at  the 
instant  considered,  it  would  pass  over  exactly  80  miles. 


I 


In  order  to  have  a  uniform  unit  of  velocity,  it  is  custom- 
ary to  express  it  in  feet  <uid  se&iiule ;  and  when  velocities 


4  ACCELBRATtOK 

are  expressed  in  any  other  terms,  they  should  be  redtieed  to 
their  equivalent  vahie  in  feet  and  seconds.  Thn  unit 
velocity,  therefore,  is  the  velocity  with  which  a  body 
describes  one  foot  in  one  second;  other  units  may  be  taken 
where  convenience  demands,  as  miles  and  hours,  etc. 

When  wo  speak  of  the  space  passed  over  by  a  body,  we 
mean  the  path  or  line  which  a  point  in  the  body  or  which  a 
particle  describes. 

7.  Fommlte  for  Velocity.— If  »  be  the  space  pa^ed 
over  by  a  particle  in  /  units  of  time,  and  v  the  velocity,  it  is 
plain  that,  for  uniform  velocity,  we  shall  have 


8 


(1) 


that  is,  we  divide  the  whole  space  passed  over  by  the  time 
of  the  motion  over  that  space. 

If  the  velocity  continually  changes,  equal  increments  are 
not  described  in  equal  times,  and  the  velocity  becomes 
a  function  of  the  time.  But  however  much  the  velocity 
changes,  it  may  be  regarded  as  constant  during  the 
infinitesimal  of  time  dt,  in  which  time  the  body  will 
describe  the  infinitesimal  of  space  ds.  Hence,  denoting  the 
velocity  at  any  instant  by  v,  we  have 


V  = 


da 
dt' 


(3) 


I- 


In  this  case  the  velocity  is  tho  ratio  of  two  infinitesimals. 
These  two  expressions  for  the  velocity  are  true  whether  the 
particle  be  moving  in  a  right,  or  in  a  curved,  line. 

8.  Acceleratioii  is  the  rate  of  change  of  velocity.  It 
is  a  velocity  increment.  If  the  velocity  is  increasing,  the 
acceleration  is  considered  positive;  if  decreasing,  it  is 
negative. 

Acceleration  is  said  to  be  uniform  when  the  velocity 


~*im 


lid  be  redtieed  to 
nds.  Thn  unit 
which  a  body 
ts  may  be  taken 
lioure,  etc. 
er  by  a  body,  we 
I  body  or  which  a 


the  space  passed 
the  velocity,  it  is 
ive 


(1) 

ver  by  the  time 

[  increments  are 
jfelocity  becomes 
ach  the  velocity 
ant  during  the 
the  body  will 
Loe,  deuotitig  the 


0  infinitesimals, 
rue  whether  the 
i,  line. 

of  vehcity.     It 

1  increasing,  the 
lecreasing,  it  is 

len  the  velocity 


MBAaURB  OF  accblbratioh.  5 

receives  eqcal  increments  in  equal  times.    Otherwise  it  is 

variable. 

9.  Measure  of  Acceleration.— Uniform  acceleration 
is  measured  by  the  actual  increase  of  velocity  in  a  unit  of 
time.  Variable  acceleration  is  measured,  at  any  instant,  by 
tlie  velocity  which  would  be  generated  in  a  unit  of  time, 
were  the  velocity  to  increase,  during  that  unit,  at  the  same 
rate  as  at  the  instant  considered. 

Calling  /the  acceleration,  v  the  velocity,  and  t  the  time, 
we  have,  when  the  acceleration  is  uniform. 


/ 


(1) 


However  variable  the  acceleration  is,  it  may  be  regarded 
as  constant  during  the  infinitesimal  of  time  dt,  in  which 
time  the  increment  of  velocity  will  be  dv.  Hence,  denoting 
the  acceleration  at  the  time  t  by/,  we  have 


(2) 


We  also  have  (Art.  8) 


V  = 


which  in  (2)  gives 


ds 
It 


-_  dw  _  rf     ds 

^  ~  dt  ~  dt  '  Jt 


d*a 


(8) 


That  is,  when  the  acceleration  is  variable  it  is  measured,  at 
any  instant,  by  the  derivative  of  the  velocity  regarded  as  a 
function  of  the  time,  or  by  the  second  derivative  of  the 
space  regarded  as.  a  function  of  the  time. 
From  (3)  we  get,  by  integration. 


da 
/*  =  rf7  =  ^' 


(♦) 


>-! 


VMLOCtTT  AND  AOCELaBATtOH, 


i/P 


rs  <; 


a  a 


HJJ 


Si" 


and 


2/8  =  v«, 


(5) 
(6) 


which  determine  the  velocity  and  space. 

10.  Gtoometrio  Reprosentatioii  of  Valoeity  and 

Acceleration.— The  velocity  of  a  body  may  be  conveni- 
ently represented  geometrically  in  magnitude  and  direction 
by  means  of  a  straight  line.  Let  the  line  be  drawn  from 
the  point  at  which  the  motion  is  considered,  and  in  the 
direction  of  motion  at  that  point  With  a  convienient  scale, 
let  a  length  of  the  line  be  cat  off  that  shall  contain  as  many 
units  of  length  as  there  are  units  in  the  velocity  to  be  repre- 
sented. The  direction  of  this  line  vnll  represent  the 
direction  of  the  motion,  and  its  length  will  represent  the 
velocity. 

Also  an  acceleration  may  be  represented  geometrically  by 
a  straight  line  drawn  in  the  direction  of  the  velocity 
generated,  and  containing  as  many  units  of  length  as  there 
are  units  of  acceleration  in  the  acceleration  considered. 
Also,  since  an  acceleration  is  measured  by  the  actual 
increase  of  velocity  in  the  unit  of  time,  the  straight  line 
which  represents  an  acceleration  in  magnitude  and  dii*ec- 
tion  will  also  completely  represent  the  velocity  genei-atcd  in 
the  unit  of  time  to  which  the  acceleration  corresponds. 

11.  The  Mass  of  a  body  or  particle  is  the  quantity  of 
matter  which  it  contains;  and  is  proportional  to  the 
Volume  and  Density  jointly.  The  Density  may  therefore 
be  defined  as  the  quantity  of  m»t>«r  in  a  unit  of  volume. 

Let  ilf  be  the  mass,  p  the  density,  and  V  the  volume,  of 
a  homogeneous  body.    Then  we  have 


M^  Vp, 


(1) 


if  we  so  take  our  units  that  the  unit  of  mass  is  the  mass  of 
the  unit  volume  of  a  body  of  unit  density. 


-■i^-VAiia:  %.  J.- 


•iv. 


▼aloeity  and 

lay  be  conveui- 
le  and  diroction 
be  drawn  from 
red,  and  in  the 
ionvenient  scale, 
contain  as  many 
Mjity  to  be  repre- 
reprosent  the 
11  represent  the 

^metrically  by 
of   the  velocity 

lengtli  aa  there 
;ion  considered. 

by  the  actual 
he  straight  line 
Itude  and  direc- 
ity  genei-ated  iu 

orrespoud*. 

the  quantity  of 
>rtional   to  the 
\y  may  therefore 
►it  of  volume, 
the  volume,  of 

(1) 

is  the  muss  of 


MOMKNTUM.  7 

If  the  density  varies  from  point  to  point  of  the  body,  we 
liave,  by  the  above  formula,  and  the  notation  of  the 
Integral  Calculus, 

3f=/odV  =  f/fpdx  dy  dt,  (2) 

wlicre  p  is  supposed  to  be  a  known  function  of  x,  y,  t. 

In  England  the  unit  of  mass  is  the  imperial  standard 
l)ound  avoirdupois,  which  is  the  weight  of  a  certain  piece  of 
platinum  preserved  at  the  standard  office  in  London.  On 
the  continent  of  Europe  the  unit  of  mass  is  the  gramme. 
This  is  Icnown  as  the  absolute  or  kinetic  unit  of  mass. 

12.  The  Quantity  of  Motion,*  or  the  Momentum 

of  a  body  moving  without  rotation  is  the  product  of  its 
mass  and  velocity.  A  double  mass,  or  a  double  velocity, 
would  correspond  to  a  double  quantity  of  motion,  and 
so  on. 

Hence,  if  we  take  as  the  unit  of  momentum  the  mo- 
mentum of  the  unit  of  mass  moving  with  the  unit  of 
velocity,  the  momentum  of  a  mass  M  moving  with  velocity 
r  is  Mv. 

13.  Change  of  Quantity  of  Motion,  or  Change  of 
Momentum,  is  proportional  to  the  mas^  moving  and  the 
change  of  its  velocity  jointly.  If  then  the  mass  remains 
constant  the  change  of  momentum  is  measured  by  the 
product  of  the  moss  into  the  change  of  velocity  ;  and  the 
rale  of  change  of  momentum,  or  acceleration  of  momentum, 
is  measured  by  the  product  of  the  mass  moving  and  the 
rate  of  change  of  velocity,  that  is,  by  the  product  if  the 
mass  moving  and  the  acceleration  (Art.  9).  Thus,  ct  lling 
J/  the  mass,  we  have  for  the  measure  of  the  rale  of  change 
of  momentum, 

rd*« 


M 


Ifl' 


*  This  phtwe  WM  a««d  bj  Newton  in  place  of  Uie  more  modern  term  "  Momen- 

mm." 


8 


STATIC   MSAaURS    OF  FORCE. 


14.  Force. — Force  is  any  cause  which  changes,  or  tends 
to  change,  a  body's  state  of  rest  or  motion. 

A  force  always  tends  to  produce  motion,  but  may  be  pre- 
vented from  actually  producing  it  by  the  counteraction  of 
an  equal  and  opposite  force.  Several  forces  may  so  act  on 
a  body  as  to  neutralize  each  other.  When  a  body  remains 
at  rest,  though  actcu  on  by  forces,  it  is  said  to  be  in 
equilibriam ;  or,  in  other  words,  the  forces  are  said  to 
produce  equilibrium. 

What  force  is,  in  its  nature,  we  do  not  know.  Forces 
are  known  to  us  only  by  their  effects.  In  order  to  measure 
them  we  must  compare  the  effects  which  they  produce 
under  the  same  circumstances. 

15.  Static  Measnre  of  Force.— 7%e  effect  of  a  force 
depends  on:  Ist,  its  magnitude,  or  intensity ;  2d,  its  direc- 
tion; i.  e.,  the  direction  in  which  it  tends  to  move  the  body 
on  which  it  acts  ;  and  3d,  its  point  of  application .;  t. «.,  the 
point  at  which  the  force  is  applied. 

The  effect  of  a  force  is  pressure,  and  may  be  expressed  by 
the  weight  which  will  counteract  it.  Every  force,  statically 
considered,  is  a  pressure,  and  hence  ha«  magnitude,  and 
may  be  measured.  A  force  may  produce  motion  or  not, 
according  as  the  body  on  which  it  acts  is  or  is  not  free  to 
move.  For  example,  take  the  case  of  a  body  restiug  on  a 
table.  The  ame  force  which  produces  pressure  on  the 
table  would  cause  the  body  to  fall  toward  the  earth  if  the 
table  were  removed. 

The  cause  of  this  pressure  or  motion  is  gravity,  or  the 
force  of  attraction  in  the  earth.  In  the  first  case  the  attrac- 
tion of  the  earth  produces  a  pressure;  in  the  second  case  it 
produces  motion.  Now  either  of  these,  viz.,  the  pressure 
which  the  body  exerts  when  at  rest,  or  the  quantity  of 
motion  it  produces  in  a  unit  of  time,  may  be  taken  as  a 
means  of  measuring  the  magnitude  of  the  force  of  attrac- 
tion that  the  earth  exerts  on  the  body.    The  former  is 


S^' 


V^'/il^^-^t,,^*i":i'«V#i4^,"^^*^.-^-S^:''~<^aSi{^S\ 


«* 


hanges,  or  tends 

but  may  be  pre- 
counteraction  of 
38  may  bo  act  on 
a  body  remains 
is  said  to  be  in 
rces  are  said  to 

t  know.  Forcea 
)rder  to  measure 
h  thoy  produce 

effect  of  a  force 
y;  2d,  its  direc- 
to  move  the  body 
ication ,;  i.  e.,  tbo 

y  be  expressed  by 
Y  force,  statically 

magnitude,  and 
B  motion  or  not, 
i  or  is  not  free  to 
>dy  resting  on  a 
pressure  od  the 

the  earth  if  the  ~ 

is  graTity,  or  the 
]t  case  the  attrac- 

e  second  case  it 
iz.,  the  pressure 

the  quantity  of 
ay  be  taken  as  a 
le  force  of  attrac- 
The  former  is 


MBTffOD    OF   COMPABiya    FOttCKS, 


called  the  static  method,  and  tUe  forces  ftrc  called  ttatie 
farces;  the  latter  is  called  the  kinetic  method,  and  the 
forces  are  called  kinetic  forces.  Weight  is  the  name  given 
to  the  pressure  which  the  attraction  of  the  earth  causes  a 
l)o(ly  to  exert  Hence,  since  static  forces  produce  pressure, 
we  may  take,  as  the  unit  of  force,  a  pressure  of  one  pound 
(Art.  11). 

Therefore,  the  magnitude  of  a  force  may  be  measured 
ainfically  hy  the  pressure  it  will  produce  "t/on  some  body, 
ami  expressed  in  pounds.  This  is  called  the  Static  measure 
of  force,  and  its  unit,  one  pound,  is  called  the  Oravitation 
unit  of  force. 

16.  Action  and  Roactioii  are  always  equal  and 

opposite. — This  is  a  law  of  nature,  and  our  knowledge  cf 
it  comes  from  experience.  If  a  force  act  on  a  body  hold  by 
a  fixed  obstacle,  the  latter  will  oppose  an  equal  and  con- 
trary rasistance.  If  the  force  act  on  a  body  free  to  move, 
motion  ¥rill  ensue ;  and,  in  the  act  of  movitig,  the  inertia 
of  the  body  will  oppose  an  equal  and  contrary  resistance. 
If  we  press  a  stone  with  the  hand,  the  stone  presses  the 
liuud  in  return.  If  we  stiike  it,  we  receive  a  blow  by  the 
act  of  giving  one.  If  we  urge  it  so  as  to  give  it  motion,  we 
lose  some  of  the  motion  which  we  should  give  to  our  limba 
In  the  same  effort,  if  the  stone  did  not  impede  them.  In 
euch  of  these  cases  there  is  a  reaction  of  the  same  kind  as 
tlic  action,  and  equal  ♦^"  it. 

17.  Method  of  Comparing  Forces.— Two  forces  are 
orjiial  when  being  applied  iu  opposite  directions  to  a 
particle  they  maintain  equilibrium.  If  we  take  two  ^qual 
forces,  and  apply  them  to  a  particle  in  the  same  direction, 
wo  obtain  a  force  double  of  either ;  if  we  unite  three  equal 
foices  we  obtain  a  triple  force ;  and  so  on.  So  that,  in 
general,  to  compare  or  measure  forces,  we  hare  only  to 
K^iopt  the  same  method  as  when  we  compare  or  measure 


JO 


REPRBSENTATIOH   OP  P0SCX8. 


any  qnantities  of  the  same  kind ;  that  is,  we  maet  take 
some  known  force  as  the  unit  of  force,  and  then  express,  in 
numbers,  the  relation  which  the  other  forces  bear  to  this 
measuring  unit.  For  example,  if  one  pound  be  the  unit  of 
force  (Art.  16),  a  force  of  12  pounds  is  expressed  by  12 ; 
and  so  on. 

18.  Rspresentatioii  of  Forces  by  SymbolB  and 
IJneg. — If  p.  Q.  B.,  etc.,  represent  forces,  they  are  numl)ers 
expressing  the  numl)er  of  times  which  the  concrete  unit  of 
force  is  contained  in  the  given  forces. 

Forces  may  be  represented  geomehrically  by  right  lines ; 
and  this  mode  of  reprci^icntation  has  the  advantage  of  giving 
the  direction,  magnitude,  and  point  of  application  of  each 
force.  Thus,  draw  a  line  in  the  direction  of  the  given 
force ;  then,  having  selected  a  unit  o2  length,  such  as  an 
inch,  a  foot,  etc.,  measure  on  this  line  as  many  units  of 
length  as  the  given  force  contains  units  of  W3ight  The 
marinittide  of  the  force  is  represented  by  the  measureti 
length  of  the  line ;  its  direction  by  the  direction  in  which 
the  line  is  drawn;  and  \t&  point  of  application  by  the  point 
from  which  the  line  is  drawn.* 

Thus,  let  the  force  Pact  at  the  point    * ^ 

A,  in    the   direction   AB,  and    let    AB  ^'■-  '• 

represent  us  many  units  of  length  as  P  contains  units  of 
force;  then  tlio  force  P  is  represented  geometrically  by 
the  line  AB ;  for  the  forti  act?  in  the  direction  from  A 
to  B ;  its  point  of  application  io  at  A,  and  its  magnitude  iii 
represented  by  the  length  of  the  line  AB. 

19.  Measure  of  Acceleratitig  Foices. — From  our 

definition  of  force  (Art.  14),  it  is  clear  that,  when  a  singk 

•  Forcof",  velocities,  and  accclnratlotiH  are  direeltd  (ptantlUfCy  p.nd  m  majr  be 
roprewntpd  by  a  line,  in  direction  and  magnitnde,  and  may  b«  coinpoand«d  In  the 
Mmu  way  a*  vtduri. 

If  auyUilus  bu  maRnHude  and  direction,  Uie  magoltode  and  dir«ctlon  taken 
togetlier  constitato  a  r«clor. 


res. 

is,  we  mast  take 
I  then  express,  in 
orces  bear  to  this 
nd  he  the  unit  of 
expreasod  by  12; 

Symbols  and 

they  are  numl)ers 
J  concrete  unit  of 

ly  by  right  lines; 
ivantage  of  giving 
pplication  of  each 
tiou  of  the  given 
ength,  ftuch  as  an 
as  many  units  of 
.  of  W3ight  The 
by  the  measureti 
ircction  in  which 
ation  by  the  point 


t    * « 

J  Fi9.  I. 

'  contains  units  of 

goonictrically  by 

direction  from  A 

its  magnitude  Ia 

xces. — From  our 

lat,  when  a  singlti 

uantUitr,  Rnd  ro  tuKj  b* 
be  compoonded  In  the 

tade  and  diroctton  Ukou 


MBASURS    OF  AOCKLSSATmO    FOSOXS. 


11 


iorce  acts  upon  a  particle,  perfectly  free  to  move,  it  must 
pioduce  motion ;  and  hence  the  force  may  be  represented 
to  us  by  the  motion  it  has  produced.  But  motion  is 
measured  in  terms  of  velocity  (Art.  7),  and  consequently  the 
\eiocity  communicated  to,  or  impressed  upon,  a  particle,  in 
a  given  time,  may  be  taken  as  a  measure  of  the  force. 
That  is,  if  the  same  particle  moves  along  a  right  line  so 
that  its  velocity  is  inorea/jed  at  a  constant  rate,  i-""  v/ill  be 
acted  upon  by  a  constant  force.  If  a  certain  uonstani  force, 
acting  for  a  second  on  a  given  particle,  generate  a  velocity 
(if  32.2  feet  per  second,  a  douWo  force,  acting  ibr  one 
second  on  the  same  particle,  would  generate  a  velocity  of 
'JiA  feet  per  second ;  a  triple  force  would  generate  a 
V  elocity  of  96.0  feet  pei*  second,  and  so  on. 

If  the  rate  of  increase  of  the  velocity,  (t.  e.,  the  accelera- 
tion), of  the  particle  is  not  uniform,  the  force  acting  on  it 
if,  not  nniform,  and  the  magnitude  of  the  force,  at  any 
imnt  of  the  particle's  path,  is  measured  by  the  acceleration 
of  the  pari.icle  at  this  point  Hence,  sine*  one  and  the 
.«imo  particle  is  capable  of  moving  with  all  possible  accelera- 
tions, all  forces  may  be  measured  by  the  velocities  they 
(jennrate  in  the  same  or  equal  particles  in  the  same  or  equal 
times.  When  forces  are  so  measured  they  are  called 
Accelerating  Forces. 

20.  Kinetic  or  Abftolute  Measnre  of  Forea.*— Let 
n  equal  particles  be  placed  side  by  side,  and  let  each  of  them 
lie  acted  on  uniformly  for  the  same  time,  by  the  same  force. 
Each  particle,  at  the  end  of  this  time,  will  have  the  same 
velocity.  Now  if  these  n  separate  particles  are  all  united  so 
as  to  form  a  body  of  n  times  the  mass  of  each  particle,  and 
If  each  one  of  them  is  still  acted  on  by  the  same  'orce  as 

*  Arta.  M,  n,  tt,  and  K,  treat  of  Um  Kinetle  moarare  of  force,  and  may  be 
omitted  till  Part  til  la  reached  ;  but  It  ii  <x)nvenlent  to  prewnt  them  once  for  all, 
»ua,  for  the  aake  of  tefereiicu  and  comparisou,  to  place  ttiein  with  the  Static 
lueaanre  of  foroo  at  the  beglnuiog  of  the  work. 


12       KINBTtO    OB   ABaOlVTS  MBA8USB   OP  tOMCS. 

before,  this  body,  at  the  end  of  the  time  oonsidered,  will 
have  U>e  sams  velocity  that  each  aeporate  partiule  had,  and 
will  \  i  cted  on  by  n  times  the  force  which  generated  this 
velocity  in  the  particle.  Comparing  a  single  particle,  then, 
with  tlxe  body  whose  maw  is  n  times  the  mass  of  this 
particle,  we  see  that,  to  produce  the  same  velocity  in  two 
bodies  by  forces  acting  on  them  for  the  same  time,  the 
maf^nitudes  of  the  forces  must  be  proportional  to  the 
masses  on  which  they  act.*  Hence,  generally,  since  force 
vaiies  as  the  velocity  when  the  mass  is  constant  (Art  10), 
and  varies  as  the  mass  when  the  velocity  is  constant,  we 
have,  by  the  ordinary  law  of  proportion,  when  both  are 
changed,  force  varies  as  the  product  of  the  mass  acted  npon 
and  the  velocity  generated  in  a  given  time  ;  that  is,  it  varies 
OS  the  quantity  of  motion  (Art  13)  it  produces  in  a  given 
mass  in  a  given  time.  If  the  force  bo  variable,  the  rate  of 
change  of  velocity  is  variable  (Art  19),  and  hence  the  force 
varies  as  the  product  of  the  cuiss  on  which  it  aots  and  the 
rtUe  of  chanyi  of  velocity,  ».  e.,  it  varies  as  the  acceleration 
of  the  momentum  (Art  U).  Therefore,  if  any  force  P  act 
on  a  mass  M,  wo  have  (Art  10) 


P<s:  2H/i 

or,  in  the  form  of  an  equation 


(1) 


(2) 


where  k  is  some  constant 

If  the  unit  of  force  be  taken  as  that  force  which,  actmg 
on  the  unil  of  mass  for  the  nnit  of  time,  generates  the  unit 
of  velocity,  then  if  Me  put  iTeaua)  to  unity,  %.».,  take  the 
unit  of  masj,  and  /equal  to  unity,  i.  «.,  take  the  nnii  of 
acceleration,  wo  mnst  have  the  force  producing  the  accol- 
enition  equal  to  the  unit  of  force,  or  P  equa'  to  unity. 


\F  roncB 

oonsidered,  will 
urtiule  had,  and 
I  generated  this 
)  particle,  then, 
B  mass  of  this 
velocity  in  two 
same  time,  the 
irtioual  to  the 
illy,  since  force 
istant  (Art  19), 
is  constant,  we 
when  both  are 
nass  acted  upoa 
that  is,  it  Taries 
ices  in  a  given 
ble,  the  rate  of 
hence  the  force 
it  acts  and  the 
the  aceeleration 
ny  force  P  act 

(1) 


(2) 

e  which,  acting 
lerates  the  unit 
t.  0.,  take  the 
ike  the  nnii;  of 
icing  the  accel- 
iqual  to  unity. 


TBB  ABSOLVTB  OK  KtNKTlC  MSASURB  OP  FORCB.   18 

Hence  k  must  also  h^  equal  to  oinity,  ^nd  we  have  the 
equation, 

P^Mf.  (8) 

Therefore,  the  Kinetic  or  Absolute  measure  of  a  force  is 
the  rate  of  change  or  acceleration*  of  momentum  it  produces 
in  a  unit  of  lime. 

If  the  force  is  constant,  (3)  booomes  by  (1)  of  Art  9, 


P  =  — ■• 
t 


w 


And  if  the  force  ii  variable,  (3)  beoomes  by  (3)  of  Art  9, 


P^M 


di^ 


w 


21.  The  AbflolQte  or  Slttetie  Thdt  of  Force.— 

A  sec  ad,  a  foot,  and  a  pound  being  the  units  of  time,  space, 
and  maw,  respectively  (Art^  7  and  12),  we  are  required  to 
find  the  corresponding  unit  pf  force  that  the  above  equation 
may  be  true.  The  unit  of  force  is  that  force  which,  acting 
for  one  second,  on  the  majs  of  one  pound,  generates  in  it  a 
velocity  of  one  foot  per  second.  Now,  from  the  results  of 
numerous  experiments,  it  has  been  ascertained  that  if  a 
body,  weighing  one  pound,  fall  froely  for  one  second  at  the 
seu  level,  it  will  acquire  a  velocity  of  about  32.2  feet  per 
second ;  t. «.,  a  force  equal  to  the  weight  of  a  pound,  if 
acting  on  the  niaaa  of  a  pound,  at  the  tea  level,  generates  in 
it  in  one  second,  if  free  to  move,  a  velocity  of  nearly  32.2 
feet  pcT  noond.     It  followa,  therefore,  that  a  f^rce  ol 

^^  of  the  weight  of  a  pound,  if  acting  on  the  mass  of 

a  pound,  at  the  lea  level,  generates  in  it  in  one  second,  if 
free  to  move,  a  velocity  of  one  foot  per  second ;  and  hence 

•  Dm  Tklt  and  MMto'i  DyMndti  or  •  Partlclt,  p. «. 


14 


MBASUBES  OF  FORCE. 


the  unit  of  force  is  ss-^  of  the  weight  of  a  pound,  or  rather 

less  than  the  weight  of  half  an  ounce  avoirdupois  ;  so  thai 
half  an  ounce,  acting  on  the  mass  of  a  pound  for  one 
second,  will  give  to  it  a  velocity  of  onw  foot  per  second. 
This  is  the  British  absolute  kinetic*  unit  of  force. 

In  order  that  Eq.  3  (Art.  80)  may  be  universally  true 
when  a  second,  a  foot,  and  a  pound  are  the  units  of  time, 
space,  and  mass  respectively,  ail  forces  must  be  expressed  in 
terms  of  this  unit 

22.  Three  Ways  of  Meaaoring  Force.— (1.)  If  a 
force  does  not  produce  motion  it  is  measured  by  the  pres- 
sure it  produces,  or  the  number  of  pounds  it  will  support 
(Art  16).  This  is  the  measure  of  Static  Force,  and  its 
unit  is  the  weight  of  a  pound. 

(2.)  If  we  consider  forces  as  always  acting  on  a  unit  of 
mass,  and  suppose  tLat  there  are  no  forces  acting  in  the 
opposite  direction,  then  these  forces  will  be  measured 
simply  by  the  velocities  or  accelerations  which  they  generate 
in  a  given  time.  This  is  the  measure  of  Accelerating  Force, 
and  its  unit  is  that  foroe  which,  acting  on  the  unit  of  mass, 
during  the  unit  of  time,  generate  the  utv",  of  velocity; 
hence  (Art  21),  the  unit  of  force  is  the  force  which,  acting 
on  one  pound  of  mass  for  one  second,  generates  a  velocity  of 
one  foot  per  second. 

(3.)  If  forces  act  on  different  masses,  and  produce  motion 
in  them,  and  we  consider  as  before  that  there  are  no  forces 
acting  in  the  opimsite  direction,  then  the  forces  are  meas- 
ured by  the  quantity  of  motion,  or  by  the  acceleration  of 
momentum  generated  ««  a  unit  of  time  (Art  20).  This  is 
the  measure  of  Moving  Force,  and  its  unit  (Art  21)  is  the 
force  which,  acting  on  one  pound  of  mass  for  one  second, 
generates  a  velocity  of  one  foot  per  second. 

*  Introduced  by  OaoM, 


JL, 


IX. 


)f  a  pound,  or  rathe 


Avoirdupois ;  so  that 
of  a  pound  for  one 
mw  foot  per  second, 
lit  of  force, 
be  universally  true 
e  the  units  of  time, 
must  be  expressed  in 

I  Force.— (1.)  If  a 
leasured  by  the  pres- 
)unds  it  will  support 
Uatic  Force,  and  its 

Etcting  on  a  unit  of 
forces  acting  in  the 
I  will  be  measured 
which  they  generate 
f  Accelerating  Force, 
on  the  unit  of  mass, 
le  un*",  of  velocity; 
!  force  which,  acting 
nerates  a  velocity  of 

and  produce  motion 
;here  are  no  forces 

he  forces  are  meas- 
the  acceleration  of 
(Art.  20).  This  is 
nit  (Art  21)  m  the 

lass  for  one  second, 


MSAmyO  OF  O  JiV  dynamicb. 


15 


It  must  be  understood  that  when  we  speak  ui  static, 
accelerating,  or  moving  forces-,  we  do  not  refer  to  different 
kinds  of  force,  but  only  to  force  as  measured  in  different 

ways. 

23.  Meaning  of  gr  in  Dynamice.— The  most  impor- 
tant case  of  a  constant,  or  very  nearly  constant,  force  is 
gravity  at  the  surface  of  the  eaxth.  The  force  of  gravity  is 
so  nearly  constant  for  places  near  the  earth's  surface,  that 
fivlling  bodies  may  be  taken  as  examples  of  motion  under  a 
constant  force.  A  stone,  let  fall  from  rest,  moves  at  first 
very  slowly.  During  the  first  tenth  of  a  second  the  velocity 
is  very  small.  In  one  second  the  stone  has  acquired  a 
velocity  of  abou*  32  feet  per  second. 

A  great  number  of  experiments  have  been  made  to  ascer- 
tain the  exact  velocity  which  a  body  would  acquire  in  one 
second  under  the  action  of  gravity,  and  freed  from  the 
resistance  of  the  air  The  most  accurate  method  is  indi- 
rect, by  meaus  of  the  pendulum.  The  result  of  pendnlum 
experiments  made  at  Leith  Fort,  by  Captain  Kater,  is, 
that  the  velocity  acquired  by  a  body  falling  unresisted  for 
one  second  is,  at  that  phuse,  32.207  feet  per  sbcond.  The 
velocity  acquired  in  one  second,  or  the  acceleration  (Art. 
10),  of  a  body  f.-'lling  freely  in  vacuo,  is  found  to  vary 
slightly  with  the  latitude,  and  also  with  the  elevation  above 
the  sea  level.  In  London  it  is  32.1889  feet  per  second.  In 
latitude  46°,  near  Bojtleaux,  it  is  32.1703  feet  per  second. 

This  acceleration  is  usually  denoted  by  g ;  and  when  we 
say  that  at  any  place  g  is  equal  to  32,  wo  mean  that  the 
velocity  generated  per  second  in  a  body  falling  freely* 
under  the  action  of  gravity  at  that  place,  is  a  velocity  of 
82  feet  per  second.  The  averafre  value  of  g  for  the  whole 
of  Great  Britain  differs  but  little  from  32.2  ;  and  hence  the 
numerical  value  of  g  for  that  country  is  taken  to  be  32.2. 

•  AbodyUi«ldtol)oino¥liig.rt'»«^wh«»ttU«ctodnponby  notwrce*  except 
tlioM  nader  eoniUenUon. 


16 


TUX  ixyiT  or  MAaa. 


The  fonnula,  deduced  from  observation,  and  a  certain 
theory  regarding  the  figure  and  density  of  the  earth,  which 
may  be  employed  to  calculate  the  moat  probable  value  of 
the  apparent  force  of  gravity,  is 

^  =  (?  (I  rf  .0061113  8in»  A), 

where  G  is  the  apparent  force  of  gravity  on  a  unit  mass  at 
the  equator,  and  g  the  force  of  gravity  in  any  latitude  A; 
the  value  of  Q,  in  terms  of  the  British  absolute  unit,  being 
32.088.     (See  Thomson  and  Tait,  p.  82^.) 

24.  Onivitatlon  Unite  of  Foroe  and  Mass.— If  in 
(8)  of  Art.  20,  we  put  for  P,  the  weight  W  of  the  body, 
and  write  g  for  /  since  we  know  the  acceleration  is  o,  (3) 
becomes  "  \  ' 

W=^mg.  (1) 


W 


W 
m  =  — . 

9 


m 


and  hence  -  may  be  taken  as  the  measure  of  the  mass. 

In  gravitation  memure  forces  are  measured  by  the  pres- 
sure they  will  prodvce,  and  the  unit  of  force  is  one  pound 
(Art.  16),  and  the  unit  of  mass  is  the  quantity  of  matter  in 
a  body  which  weighs  g  pounds  at  that  place  where  the  accel- 
eration of  gravity  is  g. 

This  definition  gives  a  unit  of  mass  which  is  constant  at 
the  same  place,  but  changes  with  the  loc  Uty ;  i  «.,  its  weight 
changes  with  the  locality  while  the  quantity  qf  matter  in  it 
remains  the  same.  Thus,  the  unit  of  mass  would  weigh  at 
Bordeaux  32.1703  pounds  (Art.  23),  while  at  Leith  Fort  it 
would  weigh  32.207  pounds.  Let  m  be  the  mass  of  a  body 
which  weighs  w  pounds.  The  quantity  of  matter  ia  |hi» 
body  remains  the  same  when  carried  from  place  to  place. 
If  it  were  possible  to  transport  it  to  another  planet  its  mass 


m,  and  a  certain 
'  the  earth,  which 
probable  value  of 

), 

>n  a  nnit  mass  at 

any  latitude  A; 

lolute  unit,  being 

id  Mass.— If  in 

W  of  the  body, 
sleration  i^  g,  (3) 


m 
m 


of  the  mass. 

^tred  by  the  pres- 
rce  is  one  pound 
'My  of  matter  in 
where  the  aceel- 

1  is  oonstant  at 
';  i^e.iiU weight 
t  of  matisr  in  it 
would  weigh  at 
iji  Leith  Fort  it 
mass  of  a  body 
'  matter  in  Ibis 
place  to  place, 
planet  its  mass 


OS  A  vrrATtON  MMAStr/fB  Of  PORCK. 


17 


would  not  be  altered,  but  its  fimght  wonM  be  very  different. 
Its  weight  wherever  placed  would  vary  directly  m  the  force 
of  gravity ;  but  the  aoeeleratioH  afeo  would  vary  diivotly  as 
tlie  force  of  gravity.  If  placed  on  the  sub,  for  example,  it 
would  weigh  about  5J8  tiraca  acr  macfe  m  ©b  the  surface  of 
the  earth  j  but  the  aeceleration  on  tht  ssn  would  also  b« 
28  times  as  much  ae  on  the  surface  of  th©  earth  ;  that  i% 
the  ratio  of  the  weight  to^  the  acceleration,  anywhere  in 

W 

the  universe  is  constant,  and  henoe    — ,  which    is  the 

9 
numerical  value  of  m  (Bq.  2),  is  oonstuit  for  tiw  same 
mass  at  all  places. 

25.  Compsiisoa    of    Gteavitatioii   and  Absolnta 

Maasnra.— The  pound  weight  has  been  long  used  for  the 
measurement  of  force  instead  of  mass,  and  i»the  recognized 
standard  of  reference.  It  came  into  general  use  because  it 
afforded  the  moat  ready  and  simple  method  of  estimating 
forces.  The  pressure  of  steam  in  a  boiler  is  always  reck- 
oned in  pounds  per  square  inch.  The  tension  of  a  string  is 
estimated  in  pounds;  the  force  necessary  to  draw  a  train  of 
cars,  or  the  pressure  of  water  against  a  look-gate,  is 
fxnivBsed  in  pounds.  Such  expressions  as  "a  force  of 
10  pounds,^"  or  "  a  pressure  of  steam  equal  to  50  pounds  on 
the  inch,"  are  of  every  day  occurrence.  Therefore  this 
method  of  measuring  forces  is  eminently  convenient  in 
practice.  For  this  reason,  and  because  it  is  the  one  used 
by  most  engineers  and  writers  of  mechanics,  we  shall  adopt 
.it  in  this  work,  and  adhere  to  the  measurement  of  force  by 
pounds,  and  give  all  our  results  in  the  usual  gravitation 
measme.    In  this  measure  it  is  convenient  to  represent  the 

W 

mass  of  a  body  weighing  W  pounds  by  the  fraction  — 

(Art  1^4),  so  that  (3)  of  Ari  20  becomes 


P  =  If 


0) 


18 


BXAMFLES. 


To  do  so  it  will  only  be  necessary  to  assnme  that  the  unit 
of  mass  is  the  quantity  of  matter  in  a  body  weighing  // 
pounds,  and  changes  in  weight  in  the  same  proportion  tbut 
g  changes  (Ai-t.  24). 

Of  course,  the  units  of  mass  and  force  in  (a)  of  Art.  30 
may  be  either  absolute  or  gravitation  units.  U  absolute, 
the  unit  of  mass  is  one  pound  (Art.  12),  and  the  unit  of 

force  is  -  pounds  (Art.  21).    If  gravitation,  the  units  arc 

g  times  as  great;  i.  «.,  the  unit  of  mass  is  g  pounds  (Art. 
24),  and  the  unit  of  foroe  is  one  pound  (Art  16). 

The  advantage  of  the  gravitation  measure  is,  it  enables  ns 
to  express  the  force  in  pounds,  and  furnishes  us  with  a  con- 
stant numerical  representative  for  the  same  quantity  of 
matter ;  that  is  to  say,  a  mass  orepresented  by  20  ou  the 
equator  would  be  represented  by  20,  at  the  pole  or  on 
the  sun.  Hence,  in  (1),  P  is  the  static  measure  of 
any  moving  force  [Art  22,  (3)],  W  is  the  weight  of  the 
body  in  pounds,  g  the  acceleration  of  gravity  (Art.  24), 

—  the-mass  upon  which  the  force  acts  [(2)  of  Art  24],  and 

which  is  frf.e  to  move  under  the  action  of  P,  the  unit  of 
mass  being  the  mass  veighing  g  pounds,  and  /  the 
acceleration  which  the  force  P  produces  in  the  mass. 

EXA.MPLBS  * 

1.  Compare  the  velocities  of  two  points  which  move 
uniformly,  one  through  5  feet  in  half  a  second,  and  the 
other  through  100  yards  in  a  minute.    Ans.  As  2  is  to  1. 

2.  Compare  the  velocities  of  two  points  which  move  uni- 
formly, one  through  720  feet  in  one  minute,  and  the  other 
through  3Jf  yards  in  three-quarters  of  a  second. 

Ans.  As  6  is  to  7. 

3.  A  railway  train  travels  100  miles  in  2  hours ;  find 
the  average  velocity  in  feet  per  second.  Ans.  73^. 


inme  that  the  unit 
i  body  weighing  // 
ae  proportion  tbut 

B  in  (3)  of  Art.  20 
mita.  If  absolute, 
i),  and  the  unit  of 

Hon,  the  units  are 

is  g  pounds  (Art. 
A.rt  16). 

are  is,  it  enables  ns 
shes  us  with  a  con- 
same  quantity  of 
ted  by  30  on  the 
at  the  pole  or  on 
static  measure  of 
the  weight  of  the 
gravity  (Art.  24), 

i)  of  Art.  24],  and 

of  P,  the  unit  of 
unds,  and  /  the 
in  the  mass. 


oints  which  move 
a  second,  and  the 
ns.  As  2  is  to  1. 

s  which  move  uni- 
iite,  and  the  other 
econd. 

ns.  As  6  is  to  7. 
in  2  hours ;  find 
Ans.  73f 


ItXAMPLBS. 


19 


4.  One  point  moves  nnjfonnly  round  the  circumference 
of  a  circle,  while  another  point  moves  uniformly  along 
the  diameter ;  compare  th^ir  velocities. 

Ahs.  As  tr  is  to  1. 

5.  Supposing  the  earth  to  be  a  sphere  26000  miles  in 
circumference,  and  turning  round  once  in  a  day,  deter- 
mine the  velocity  of  a  point  at  the  equator. 

Ans.  1527|  ft.  per  sec. 

6.  A  body  has  described  60  feet  from  rest  in  2  second^ 
with  uniform  acceleration ;  find  the  velocity  acquired. 

From  (1)  of  Art.  9  we  have 

and  from  (4)  we  have     /<  =  v ; 
.'.    V  =  60. 

7.  Find  the  time  it  will  take  the  body  in  the  last  exam- 
ple to  move  over  the  next  160  feet. 


From  (6)  of  Art.  9  we  have 


etc 


Ans.  2  seconds. 


8.  A  body,  moving  with  uniform  acceleration,  describes 
63  feet  in  the  fourth  second ;  find  the  acceleration. 

Ans.  18. 

9.  A  body,  with  uniform  acceleration,  described  72  feet 
while  its  velocity  increases  irom  16  to  20  feet  per  second ; 
find  the  whole  time  of  motion,  and  the  acceleration. 

Ans.  20  seconds ;  1. 

10.  A  body,  in  passing  over  9  feet  with  uniform  accelera- 
tion, has  its  velocity  increased  from  4  to  5  fe  *:  per  second ; 
find  the  whole  space  described  from  rcsut,  and  thie  accelera- 
tion. Ans.  26  feet ;  |. 


20 


MXAMPLMa. 


11.  A  body,  aniformly  accelerated,  is  found  to  be  mov- 
ing at  the  end  of  10  seconds  with  a  velocity  which,  if 
continued  uniformly,  would  caiJ7  it  through  46  mites  in 
the  next  hour ;  find  the  acceleration.  Am.  ^. 

12.  Find  the  matis  of  a  straight  wire  or  rod,  the  d^naity 
of  which  varies  directly  as  the  distance  fh>m  one  end. 

Take  the  end  of  the  rod  aa  origin  ;  let  o  =  its  length ; 
let  the  ''aetance  of  ary  point  of  it  from  that  end  =  a; ; 
and  let  u  =  the  area  of  its  transverse  section,  and  k  =  the 
density  at  the  uxut'g  distance  from  the  origin.    Thea 

dV=i<adx',    and    p=zhx', 

and  (2)  of  Art.  11  becomes 


M 


2 


13.  J?iiwl  aie  ma«i  of  a  circular  plate  of  uniform  thick- 
ness, the  density  of  whiph  varies  as  the  distance  from  the 
centre. 

Am.  |7r*Aa«,  where  a  is  the  radius,  k  the  density  at 
the  unit's  distance,  and  h  the  thickness. 

14.  Find  the  mass  of  a  sphere,  whose  density  varies 
inversely  as  the  distance  flrom  the  centre. 

An3.  2jrp«»,  where  p  is  thje  density  of  the  onteide  strfttooik 


ind  to  be  mov- 
locity  which,  if 
igh  45  mileg  ia 

rod,  the  d^wity 
1  ooe  end. 

(  =  its  length  ; 
that  end  =  x ; 
a,  and  k  =  the 
n.    Thea 


uniform  thick- 
tanoe  from  tho 

the  density  at 

density  y^es 
•ntside  stratoo^ 


STATICS     (REST) 


CHAPTER    U. 

THE    COMPOSITION    AND    RESOLUTION    OF  CONCUR. 
RING    FORCES-CONDITIONS    OF    EgUILIBRIUM. 

26.  Probimp  of  Staftios.  — The  primary  conception  of 
force  is  that  of  a  caoae  of  motimi  (Art.  14).  If  only  one 
force  acts  on  a  particle  it  is  clear  that  the  particle  cannot 
remain  at  rest.  In  statics  it  is  only  the  tendmey  which 
for  ,$8  have  to  prodaoe  motion  that  is  considered.  There 
must  be  at  least  two  forces  in  statics ;  and  they  Me  con- 
sidered as  acting  so  as  to  counteract  each  other^t  tefide^oy 
to  cause  motion,  thereby  producing  a  state  of  equilibriara 
in  the  bodies  to  which  they  are  applied.  The  ftjrces  which 
act  upon  a  body  may  be  in  equilibrium,  and  yet  motion 
exist;  but  ia  such  cases  the  motion  is  uniform.  Hence 
there  are  two  kinds  of  equilibrium,  the  one  relating  to 
bodies  at  rest,  the  other  relating  to  bodies  in  motion.  The 
former  is  sometimes  called  Static  Equilibriam  and  the  lat- 
ter Kinetic  (or  Dynamic*)  Equilibrium.  3%e  problem  of 
atatica  ia  to  detei^ine  the  conditiona  under  vohick  foreea  act 
when  they  keep  bodiea  at  rest. 

27.  Cdncnrring  and  Coiwpiring  Forces.— Resnlt- 

ant.— When  sereral  forces  have  a  common  ryoint  of  applir 
cation  they  are  called  concurring  forces ;  whan  they  act  ivt 
the  same  point  and  along  the  same  right  line  they  are 
called  contpirinff  forces. 

The  resultant  of  two  or  more  forces  is  that  force  which 
singly  wiU  produce  the  same  effect  as  the  forces  them- 
selves when  acting  together.  The  individual  foroec^  when 
considered  with  reference  to   this   resultant,  are   called 

"— ■""    ™^™^^— — ^W*^™^— ^Wl      ■"■        ■■-     I    I  11  I  11,  MHi      ■■II. PM IIHI     I  H"l 

*  Oregoiy's  XecbMilG*,  p.  11 


m 


coMPoamoN  of  conspibinq  porcbs. 


components.    The  process  of  finding  the  resultant  of  several 
forces  is  called  the  composition  of  forces. 

28.  Composition  of  Conapiring  Forces.— Condi- 
tion of  Equilibrium. — When  two  or  more  conspiring 
forces  act  in  the  same  direction,  it  h  evident  that  the 
resaltant  force  is  equal  to  their  sum,  and  acts  m  the  same 
direction. 

When  two  conspiring  forces  act  in  opposite  directions 
their  resultant  force  is  equal  to  their  difference,  and  acts  in 
the  direction  of  the  greater  component 

When  several  conspiring  forces  act  in  different  directions 
the  resultant  of  the  forces  acting  in  one  direction  equals 
the  sum  of  these  forces,  and  acts  in  the  same  direction ; 
and  so  of  the  forces  acting  in  the  opposite  direction. 
Therefore,  the  resultant  of  all  the  forces  is  equal  to  the 
differepce  of  these  sums,  and  acts  in  the  direction  of  the 
greater  sum.  Hence,  if  the  forces  acting  in  one  direction 
are  reckoned  positive,  and  those  in  the  opposite  direction 
negative,  their  resultant  is  equal  to  their  algebraic  sum ; 
its  sign  determining  the  direction  in  which  it  acts.  Thus, 
if  Pit  -Pg,  'Pi,  etc.,  are  the  conspiring  forces,  some  of 
which  may  be  positive  and  the  others  negative,  and  R  is 
the  resultant,  we  have 


/?  =  P,  +  P,  +  P,  +  eta  =  £P, 


(1) 


in  which  S  denotes  the  algebraic  sum  of  the  terms  similar 
to  that  written  immediately  after  it« 

CoE.— The  condition  that  the  forces  may  be  in  equilib- 
rium is  that  their  resultant,  and  therefore  their  algebraic 
sum,  must  vanish.  Hence,  when  the  forces  are  in  equilib- 
rium we  must  have  i?  =  0 ;  therefore  (l)  becomes 


■Pi  +  ^1  +  Pi  +  etc  =  £P  =  0. 


(») 


it  of  several 


I.— Condi- 
conspiring 
tt  that  the 
I  the  same 

directions 
ind  acts  in 

directions 
bion  equals 
direction ; 
direction, 
nal  to  the 
ion  of  the 
9  direction 
i  direction 
raic  sum; 
ts.  Thns, 
,  some  of 
and  Ji  is 


(1) 
as  similar 


tt  ©qnilib- 

algebraie 

D  eqoilib- 


(2) 


COMPOSITION  OF  VSL0CITIS8.  88 

t 

29.  Composition  of  VelocltieB.— //  »  particle  be 
moving  with  two  uniform  velocities  represented  in 
magnitude  and  direction  by  the  two  adjacent  sides 
of  a  parallelogram,  the  resultant  velo'  Uy  will  be 
represented  in  majnitudi  and  direction  by  the 
diagonal  of  the  parallelogram. 

Let  the  particle  move  with  a  uniform 
velocity  t',  which  acting  alor:3  will  take 
it  in  one  i  :cond  from  A  to  B,  and  with 
a  uniform  velocity  v',  which  acting 
alone  will  take  it  in  one  second  from  A 
to  C  ;  at  the  end  of  one  second  the  par- 
ticle will  be  found  at  D,  and  AD  will  represent  in  magni- 
tude and  direction  the  resultant  of  the  velocities  represented 

by  AB  and  AC. 

Suppose  the  particle  to  r^oro  uniformly  along  a  straight 
tube  which  starts  from  AB,  and  moves  uniformly  parallel 
to  itself  with  its  extremity  in  AC.  When  the  particle  stary 
from  A  the  tube  is  in  the  position  AB.  When  the  particle 
has  moved  over  any  part  of  AB,  the  end  of  the  tube  has 
moved  oyer  the  same  part  of  AC,  and  the  particle  is  on  the 

line  AD.    For  example,  let  AM  be  the  -  th  part  of  AB,  and 

AN  be  the  -th  part  of  AC  ;  while  the  particle  moves  from 

n 
A  to  M,  the  end  A  with  the  tube  AB  will  move  from  A  to 
N,  and  the  particle  will  be  at  P,  the  tube  occupying  the 
position  K  L,  and  PM  being  parallel  and  equal  to  AN.    F 
can  be  proved  to  be  on  the  diagonal  AD  as  follows : 


AM  :  MP   : :   — 


AB     AC 


n 


AB  :  AC  (=  BD); 


therefore  P  lies  on  the  diagonal  AD.    Also  since 
AM  :  AB   : :    AP  :  AD, 


fc»W*W.pi»««i*^»"*'^*'  ■ 


84 


coMPoamoH  ow  mnomiL 


the  resultsnt  velocity  is  unifonn.  Hence,  the  diagonal  AD 
represents  iu  magnitude  and  diniotioa  the  reaaltunt  of  the 
Telocities  represented  by  AB  and  AC. 

This   proposition   is   known   as  th0  Parallelogram  of 
Vehciiies. 


30.  Compositloii  of  Forces.— From  the  Parallelo- 
gram of  Velocities  the  Parallelogram  of  Forces  follows 
immediately.  Since  two  simultaneous  velocities,  AB  and 
AC,  of  a  particle,  result  in  a  single  velocity,  AD,  nnd  since 
these  three  velocities  may  bo  regarded  as  the  measures  of 
three  separate  forces  all  acting  for  the  same  time  (Art.  10), 
it  follows  that  the  cifect  produced  on  a  particle  by  the  com- 
bined action,  for  the  same  time,  of  two  forces  may  be  pro- 
duced by  the  action,  for  the  same  time,  of  a  single  force, 
which  is  therefore  called  the  resultant  of  the  other  two 
forces ;  and  these  forces  ore  represented  in  magnitude  and 
direction  by  AB,  AC,  and  AD.  (See  Minchin,  p.  7,  also 
GametfB  Dynamics,  p.  10.) 

Hence  if  two  concurring  farces  be  represmted  in  magni- 
tude and  direction  by  the  adjacent  sides  of  a  parallelogram, 
their  resultant  mil  be  represented  in  mag7iitude  and  direction 
by  the  diagonal  of  the  paraiielogram.  Care  must  be  token 
in  oonstructiug  the  parallelogram  of  forces  that  the  com- 
ponents both  aot  from  the  angle  of  the  parallelogram  iVom 
which  the  diagonal  is  drawn. 

Thii  propoaitioB  haa  bsen  proved  in  vmtIoim  wkju.  It  wm  ennn- 
dated  !a  :t«  pnweDt  form  by  Sir  laMO  Nomumi,  and  bjr  Varisuon,  the 
oolebnted  mathematiciHn,  In  tlie  jmu.*  1067,  prcbablj  independent  of 
each  other.  Since  that  time  vartoiu  proofe  of  it  have  been  given  by 
difibrent  mathematicians.  One  wcdtk  gives  a  dlacaiBion,  mora  or  leoa 
complete,  of  4lt  other  proofs.  A  noted  analytic  proof  is  given  by 
M.  PnisBon.  (See  Prine'e  CaL,  Vol.  Ill,  p.  19).  Some  authors  object 
to  proving  the  parallelogram  of  forces  by  means  of  the  parallelogram 
of  velocities.  (See  Gregory's  Meehanlos,  p.  14)  The  student  who 
wants  other  proofs  is  referred  to  Dochayla's  proof  ss  found  in  Tod- 
hunter's  Statics,  p.  7,  and  in  Ualbraitb's  MeohsAios,  p.  7,  and  in  many 


)  diagonal  AD 
saltuDt  of  the 

iHehgram  of 


tie  Paralleh- 
'oreea  follows 
ties,  AB  and 
tD,  and  since 

measures  of 
ne  (Art.  10), 

by  the  corn- 
may  be  pro- 
single  force, 
e  other  two 
gnituda  and 
3>  p.  7,  also 

tl  in  magni- 
ralUhgrmn, 
'nd  direction 
«st  be  taken 
»t  the  com- 
ogram  fcom 


It  WM  «nun- 
t'^ariguon,  the 
dependent  of 
Ben  given  by 
,  more  or  less 

1»  given  by 
ithon  object 
•nllelogram 
student  who 
>and  in  Tod- 
ftnd  in  niaoy 


TMiANOhE  or  woacsa. 

other  works ;  or  to  UpUoe-s  pnxrf,    (Sm  M^canicine  Celeste,  Liv.  I, 
chap.  1.) 

If  fl  be  the  angle  between  the  sides  of  the  parallelogram, 
AJi  and  AC  (Fig.  2),  and  P  and  Q  represent  the  two  com- 
ponent forces  acting  at  A,  and  R  represent  the  resultant, 
AD,  we  have  from  trigonometry, 


iZ»  =  i«  -f.  ^  -H  ^PQ  cos  9 


m 


an  equation  which  gives  the  magnitude  of  the  resultant  of 
two  forf  es  in  terms  of  the  magnitudes  of  the  two  forces  and 
the  angle  between  their  directions,  the  forces  being  repre- 
sented by  two  lines,  both  dra\ni  from  the  point  at  which 
they  act. 

Cob. —If  e  =  90^  and  «  and  0  be  the  angles  which  the 
direction  of  R  makes  with  the  directions  of  P  and  Q,  we 
have  from  (1) 


C0B«  = 
COS/3  =  ^; 


(») 


(K) 


from  which  the  magnitude  and  direction  of  the  resultant 
are  determined. 

31.  Triangte  of  Tonm.—If  three  oonourring 
forces  be  repreaenied  in  magnitude  and  direoiicn 
hij  the  sides  of  a  triangle,  taken  in  order,  they  will 
be  in  equilibrium. 

Let  ABO  be  the   triangle  who«" 
sides,  taken  in  order,  represent  in 
magnitude  and  direction  three  foroes 
applied  at  the  point  A.    Complete 
% 


n*>s 


S6 


TBlAJiGLM  OF  FOBCXS. 


the  pawllelogram  ABCD.  Then  the  forces,  AB  and  EC, 
applied  at  A,  are  expressed  by  AB  and  AD  (since  AD  is 
e<iual  and  parallel  to  BC).  But  the  resultant  of  AB  and 
AD  is  AC,  acting  in  the  direction  AC.  Therefore  the  three 
forces  represented  by  AB,  BC,  and  CA,  are  equivalent  to 
two  forces,  AC  and  CA,  the  fomer  acting  from  A  towards 
C  and  the  latter  from  C  towards  A,  which,  being  equal  and 
opposite,  will  clearly  balance  each  other.  Therefore  the 
three  forces  represented  by  AB,  BC,  and  CA,  acting  at  the 
point  A,  will  be  in  equilibrium. 

It  should  be  observed  that  though  BC  represents  the 
magnitude  and  direction  of  the  component,  it  is  not  in  the 
Une  of  its  action,  becanse  the  three  forces  act  at  the 
♦    ut  A. 

The  converse  of  this  is  also  true  ;  viz..  If  three  concurrinf^ 
forces  are  in  equilibrium,  they  may  be  represent/nl  in  m!;g- 
nitude  and  direction  by  the  sides  of  a  triangle,  drawn 
parallel  respectively  to  the  directions  of  the  forces. 

Thus,  if  AB  and  BC  represent  two  forces  in  magnitude 
and  direction,  AC  will  represent  the  resultant,  and  henre  to 
produce  equilibrium  tbe  resultant  force  AC  must  be  opposed 
by  an  equal  and  oppoi  ite  force  CA.  Therefore,  the  three 
forces  in  equilibrium  will  be  represented  by  AB,  BC,  and 
CA. 

CoE. — When  three  concurring  forces  are  in  equilibrium, 
each  is  equal  and  directly  opposite  to  the  resultant  of  the 
other  tw',  • 

it.  1«>J:^«<  .ns  iMtwosn  ThrM  Oonewring  ForoM 

in  Tic  i^■  vm.— Siuce  the  sides  of  a  plane  triangle  are 
as  the  Biuo'   >/  iie  opposite  angles,  wo  have  (Fig.  8) 

AB  :  BC  (or  AD) :  AC  : :  sin  ACB  :  sin  BAC  :  sin  ABC 

: :  sin  DAC  :  sin  BAC  :  sin  BAD. 

Hence,  calling  P,  Q,  and  R,  the  forces  represented  by  AB, 
AD,  and  AC,  and  denoting  the  angles  between  the  direo- 


lu. 


mm 


!8,ABand  BC, 
D  (since  AD  is 
tant  of  AB  and 
irefore  the  three 
e  eqaivaleut  to 
Tom  A  towards 
being  equal  and 
Therefore  t!ie 
i,  acting  at  the 

represents  the 
it  is  not  in  the 
ces  act  at  the 

hree  concnrrin/-j 
sent/nl  in  m^g- 
triangle,  drawn 
forces. 

8  in  magnitude 
it,  and  henre  to 
nust  be  opposed 
(fore,  the  three 
)y  AB,  BO,  and 

in  equilibrium, 
reeuluuit  of  the 


irrinf  ForoM 

me  triangle  are 
(Fig.  8) 


AC: 
AC 


sin  ABO 
sin  BAD. 


aseuted  by  AB, 
woen  the  direo- 


POLTQON   OP  FORCSa. 


tions  of  the  forces  P  and  Q,  Q  and  R,  and  R  and  P,  by 
AAA 
PQ>  QP>  and  RP,  respectively,  we  have 


P      ^  _Q R 

.     /^  /\  A 

em  QR       sin  RP       sin  PQ 


0) 


There/ore,  when  three  concurring  forces  are  in  equilibrium 
they  are  respectively  in  the  same  proportion  as  the  sines  of 
the  angles  included  between  the  directions  of  the  other  two. 

33.  The  Polygon  of  Forces.— //  amj  number  of 
concurring  forces  be  represented  in  magnitude  and. 
direction  by  the  sides  of  a  closed  polygon  taken  in 
o:  der,  they  will  be  in  equilibriurrv. 

Let  the  forces  be  represented  in  It 
magnitude  and  direction  by  the  lines 
AP„  AP„  AP„' AP„  AP,.  Take 
AB  to  represent  AP,,  through  B  draw 
BC  equal  and  parallel  to  AP,  ;  the 
resultant  of  the  forces  AB  and  BC,  or 
AP,  and  AP,  is  represented  bj  AO 
(Art.  31).  Of  courae  the  resultant 
acts  at  A  and  is  parallel  to  BC.  Again  through  0  draw  CD 
equal  and  parallel  to  AP„  the  resultant  of  AC  and  CD,  or 
AP,,  AP„  and  AP,  is  AD.  Also  through  D  draw  DE 
equal  and  parallel  to  AP^,  the  resultant  of  AD  and  DE,  or 
AP,,  APg,  APj,  and  AP^  is  AE.  Now  if  AE  is  equal  and 
opposite  to  AP,  the  system  is  in  equilibrium  (Art.  18). 
Hence  the  fos-ces  represented  by  AB,  BC,  CD,  DE,  EA 
wiU  be  in  equilibrium. 

Cob,  1. — Any  one  side  of  the  polygon  represents  in 
magnitude  and  direction  the  resultant  of  all  *he  forces 
represented  by  the  remaining  sides. 

Cob.  2.— If  the  lines  representing  the  forces  do  not  form 
a  closed  polygon  the  forces  are  not  in  eqoilibrium  ;  in  this 


»8 


PARALLMI'OPIPED    OP  P0K0S8. 


case  the  last  side,  AE,  taken  from  A  to  E,  or  that  which  is 
required  to  clope  up  the  polygon,  represents  in  magnitude 
and  direction  the  resultant  of  the  system. 

34.  Farallelopijied  of  Porcea.— //  three  concur- 
ring forces,  not  ih.  the  sattve  plane,  are  represented 
in  magnitude  and  direction  by  the  three  edges  of 
a  parallelopiped,  then  the  resultant  will  be  repre- 
sented in  magnitude  and  direction  by  the  diag- 
onal; conversely,  if  the  diagonal  of  a  parallel- 
opipcd  represe.its  a  force,  it  is  equivalent  to  three 
forces  represented  by  the  edges  of  the  parallci- 
opiped. 

Let  the  three  edges  AB,  AC,  AD  of  the 
parallolopiped  represent  the  three  forces, 
applied  at  A.  Then  the  resultant  of  the 
forces  AB  and  AC  is  AE,  the  diagonal  of 
the  face  ABCE;  and  the  resultant  of  the 
for'^os  4^  and  AD  is  AF,  the  diagonal  of 
the  parallelogram  ADFE.  Hence  AF  represents  the 
resultant  of  the  three  forces  AB,  AC,  and  AD. 

Conversely,  the  force,  AF,  is  equivalent  to  the  three 
components  AB,  AC,  and  AD. 

Lot  F,  Q,  S  represent  the  three  forces  AB,  AC,  AD ;  R, 
the  resultant ;  a,  P,  y,  the  angles  whicL  the  direction  of  B 
makes  with  the  directions  of  P,  Q,  S,  and  suppose  the 
forces  to  act  at  right  angles  with  each  other.    Then  aiuce 

Sr»  =  XB*  +  AC»  +  AD', 

we  have  R^  =  P*  +  (?  +  3*; 

,  P 

alio,  cos  o  =  -gi 

Q 
cos  /3  =  ^, 

8 


(1) 


m 


ta, 

\T  that  which  is 
I  in  magnitude 

Ihree  concur- 
s  represented 
tree  edges  of 
vill  be  repre- 
by  the  diag- 
n  parullel- 
Zent  to  three 
'ihe  parallci- 


represents  the 

D. 

;  to  the  three 

J,  AC,  AD;  R, 

direction  of  R 

kd  suppose  the 

Then  aiuce 


(1) 


(a) 


KXAMPtiiS. 


»» 


from  which  the  magnitude  and  direction  of  the  resultant 
are  determined. 

EXAMPLES. 

1.  Three  forces  of  5  lbs.,  3  lbs.,  and  2  lbs.,  respectively, 
act  upon  a  point  in  the  same  direction,  and  two  other  forces 
of  8  lbs.  and  9  lbs.  act  in  the  opposite  direction.  What 
single  force  will  keep  the  point  at  rest  P  Atts.  7'lbs. 

a.  Two  forces  of  5|  lbs.  and  3|  lbs.,  applied  at  a  point, 
urge  it  in  one  dire^'^ion ;  and  a  force  of  2  lbs.,  applied  at 
the  same  point,  urges  it  in  the  opposite  direction.  What 
additional  force  is  necessary  to  preserve  equilibrium  ? 

Ans.  t  lbs. 

3.  If  a  force  of  13  lbs.  be  represented  by  a  line  of  6^ 
inches,  what  line  nWX  represent  a  force  of  7^  lbs.? 

Am.  8f  inches. 

4.  Two  forces  whose  magnitudes  are  as  3  to  4,  acting  on 
a  point  at  right  angles  to  each  other,  produce  a  rcuultaut  of 
20  lbs.;  required  the  component  forces. 

Am.  12  lbs.  and  16  lbs. 

5.  Let  ABO  be  a  triangle,  and  D  the  middle  point  of 
the  side  BC.  If  the  three  forces  represented  in  magnitude 
and  direction  by  AB,  AO,  and  AD,  act  upon  the  point  A ; 
And  the  direction  and  magnitude  of  the  rennltant 

Ans.  The  direction  is  in  the  line  AD,  and  the  magni- 
tude is  represented  by  SAD. 

6.  When  P  =  Q  and  fi  =  60°,  find  R. 

Ana.  R  =  PVS. 

7.  When  P  =  Q  and  8  =  135°,  find  R. 


Ans.   R-  P^'z  —  V^ 

8.   When  P  =  ^  and  0  =  120°,  find  R. 

Ans.  R  =  P. 


30 


sssoLurrox  or  roscss. 


\ 


'■ 


9.  If  P  =:  Q,  show  that  their  resultant  R  —  ^P  cos  ?• 

10.  If  P  =  8,  and  e  =  10,  and  (?  =  60°,  find  R. 

Ana.  R  z=2  V^l. 
'll.  If  P  =  144,  R  =  145,  and  6  =  90°,  find  Q. 

Ans.  Q  =  17. 

12.  Two  forces  of  4  lbs.  and  3  V2  lbs.  act  at  an  angle  of 
45°,  and  a  third  force  of  V42  lbs.  acts  at  right  angles  to 
their  plane  at  the  same  point ;  find  their  resultant. 

Ahs.  10  lbs. 

35.  Resolntion  of  Forces.— %  the  resohition  of  forces 
is  meant  the  process  of  finding  tlie  components  of  given  forces. 
We  have  seen  (Art  30)  that  two  concurring  forces,  P  and 
C  =  AB  and  AC,  (Pig.  2)  are  equivalent  to  a  single  force 
72  =  AD ;  it  is  evident  then  that  the  single  force,  R,  ucnng 
along  AD,  can  be  replaced  by  the  two  forces,  P  and  Q, 
represented  in  magnitude  and  direction  by  two  adjacent 
sides  of  a  parallelogram,  of  which  AD  is  the  diagonal. 

Since  an  infinite  nnmber  of  pamllelogrnnis,  of  oac'i  of 
which  AD  is  the  diagonal,  can  be  constructed,  it  follows 
that  a  single  force,  R,  can  be  resolved  into  two  other  forces 
iu  an  infinite  number  of  ways. 

Also,  efl 3h  of  i-ie  forces  AB,  AC,  may  be  resolved  into 
two  others  in  way  similar  to  that  by  which  ID  was 
resolved  .nto  twt ;  and  so  on  to  any  extent.  Hence,  a  single 
force  nray  be  resolved  into  any  number  of  forces,  whose 
oombinod  action  is  equivalent  to  the  original  force. 

Cor. — The  most  convenient  compo- 
nent into  which  a  force  can  be  resolved 
are  those  whose  directions  are  at  right 
angles  to  eacli  other.  Thus,  let  OX 
and  OF  be  any  two  lines  at  right 
ujiglea  to  each  other,  and   P  any  force  acting  at  0  in  the 


Fia.a 


i  =  2P  cos  J 

9°,  find  R. 

E  —  2  V'ai. 

\  find  Q. 
ns.  Q  =  17. 

at  an  angle  of 

right  angles  to 
111  taut. 
Atis.  10  lbs. 

)hition  of  forces 
of  given  forces, 
l  forces,  P  and 
a  single  force 
force,  it,  acting 
rcos,  P  and  Q, 
'  two  adjacent 
diagonal, 
ms,  of  oac'i  of 
:ted,  it  follows 
wo  other  forces 

e  resolved  into 
vhich  AD  was 
Hence,  a  single 
forces,  whose 
force. 


a 


ig  at  0  in  the 


MAomrmiS  and  DinEcrtoir  or  bssultant. 

plane  XOY.  Then  completing  the  rectangle  OMPN  we 
find  the  ocnponenta  of  P  along  the  axes  OX  and  OF  to  be 
Oif  and  ON,  which  denote  by  X  and  Y.  Then  we  have 
clearly 

X  =  P  cos  «,  )  /jv 

r  =  P  sin  «; )  ^  ' 

where  a  is  the  angle  which  the  direction  of  P  makes  with 
OX.     These  wmponents  X  and  Y  are  called  the  rect- 
angular components.     The  i-ectangular  component  of  a 
force,  P,  along  a  right  lino  is  Px cosine  of  angle  between^ 
line  and  direction  of  P. 

In  strictness,  when  we  speak  of  the  component  of  a  given 
force  along  a  certain  line,  it  is  necessary  to  mention  the 
other  line  along  which  the  other  component  acts.  In  thia 
work,  unless  otherwise  expressed,  the  component  of  a  force 
along  any  line  will  be  understood  to-be  its  rectangular 
component;  le.,  the  resolution  will  be  made  along  this  line 
and  the  line  perpendicular  to  it. 

36.  To  find  th«  lAagnitode  and  Direction  of  the 
Resultant  of  any  number  of  Concurring  Forces  in 
one  Plane.— When  there  are  several  concurring  forces,  the 
condition  of  their  equilibrium  may  be  expressed  as  in 
Art.  33,  Cors.  1  and  2.  But  in  practice  we  obtain  much 
simpler  results  by  using  the  principle  of  the  Resolutwn  of 
Forces  (Art  -35),  than  those  given  by  the  principle  of 
Composition  of  Forces. 

Let  0  be  the  point  at  which  all 
the  forces  act.  Through  0  draw  the 
rectangular  axes  XX',  YY'.  I^t 
P,,  P„  P,,  etc.,  be  the  forces  and 
«,,  «„  a  J,  etc.,  be  the  angles  which 
their  directions  make  with  the  axis 

of*.    • 

Now  resolve  each  force  into  its  two 
components  along  the  axes  of  x  and 


Fig.7 

Then  the  com- 


88    MAomruDJi  avd  otRscnoN  of  sxauLTAjrr. 


f 


L 


ponenta  along  the  axia  of  x  (aHwmponents)  ore  (Art 
35,  Cor.),  P,  COS  a,,  P,  cos  a,,  P^  cos  «,,  etc.,  and  thoae 
along  the  axis  of  y  are  Pj  sin  ct|,  P,  sin  «„  P,  ain  «„ 
etc. ;  and  therefore  if  X  and  Y  denote  the  algebraic  sum  of 
the  ^-components  and  ^-components  respectively,  we  have 

X=  P,  cos  cr^+P,  cos  «,4-P,  cosa,  4-etc. )      . 
=  2P  coa  «,  )     ^  ' 


y  =  P,  sin  cci  +  Pi  sin  a,  +  Pj  ain  a,  +etc. 
=r  iP  sin  a. 


(2) 


Let  R  be  the  resultant  of  all  the  forces  acting  at  0,  and  0 
the  angle  which  it  makes  with  the  axis  of  x ;  then  resolving 
R  into  its  x-  and  ^-components,  we  have 


i2  COS  e  =  -T  =  £P  coa  o, 
5  si-  3  3=  y  =s  £P  ain  a, 


'} 


/?  =  X»+r»;  tane  =  -=. 


(8) 


which   determines  the  magnitude  and  direction  of  the 
resultant. 

ScH. — Regarding  OX  and  OF  as  positive  and  OX^  and 
OY^  as  negative  as  in  Anal.  Geom.,  we  see  that  Oa;,,  Oy^, 
Oy,  are  positive,  and  Oa;,,  Oa;„  Oy,  are  negative.  The 
forces  may  always  be  considered  as  positive,  and  hence  the 
signs  of  the  components  in  (1)  and  (2)  will  be  the  same  as 
those  of  the  trigonometric  functions.  Thus,  since  a,  is 
>  90°  and  <  180°  its  sine  is  positive  and  cosine  is  negative; 
since  a,  is  >  180°  and  <  270°  both  its  sine  and  cosine  are 
negative. 

37.  Tha  Conditions  of  Eqnlllbrtam  for  any  nnmbor 
of  ConcTuring  ForcM  in  one  Plane.— For  the  equilibrinm 
of  the  forces  we  must  have  R  =  0.  Hence  (4)  of  Art.  36 
becomes 

X»  -H  r»  =  0.  (I) 


Urn 


VLVAHT, 

s)  ore  (Art 
3tc.,  and  thoae 
Kg,  P,  ain  «„ 
robraic  sum  of 
ely,  we  have 


4- eta) 


(1) 


+eto. 


}     <^\ 


g  at  0,  and  9 
then  resolving 


(3) 

■',  (4) 

lOtion  of  the 

and  OX^  and 
hat  Oa;,,  Oy„ 
jgative.  The 
md  hence  the 
)  the  same  as 
B,  since  a,  is 
ae  is  negative; 
nd  cosine  are 

•nynnmbcr 

tie  eqnilibrinm 
4)  of  Art  36 

(1) 


BXAMPLMB. 


88 


Now  (1)  cannot  be  satigfled  so  long  'as  X  and  F  are  real 
quantities  unleae  X=:0,  F.=:a;  therefore, 

X=:£i'co8«  =  0  and  F=SPsino  =  0.       (2) 

Hence  these  are  the  two  necessary  and  sufficient  conditions 
for  the  equilibrium  of  the  forces;  that  is,  the  algebraic  sum 
of  the  rectangular  components  of  the  forces,  along  each  of 
two  right  lines  at  right  angles  to  each  other,  in  the  plane  of 
the  forces,  is  eqwA  to  zero.  As  the  conditions  of  equilibrium 
must  be  independent  of  the  system  of  co-ordinate  axes,  it 
follows  that,  if  any  number  of  concurring  forces  in  one 
plane  are  in  equilibrium,  the  algebraic  sum  of  the  rectan- 
gular components  of  the  forces  along  every  right  line  in  their 
plane  is  zero* 

EXAMPLES. 

1.  Given  four  equal  concurring  forces  whose  directions 
are  inclined  to  the  axis  of  z  at  angles  of  16°,  76°,  136% 
and  225° ;  determine  the  magnitude  and  direction  of  their 
resultant 

Let  each  force  be  equal  *  >  P ;  then 

X  =  P  ooa  16°  +  P  coe  76   +  P  cos  136°  +  P  cos  225° 

.3* -a 


=  p: 


2* 


F  =  P  sin  16°  +  P  sin  76°  +  P  sin  136°  +  P  sin  226" 

.'.    7?  =  P(6-2V^)*- 
3* 


tan  ds= 


8 


Err 


2 


2.  Giren  two  equal  concurring  forces,  P,  whose  direc- 
f  ions  are  inclined  to  the  axis  of  x  at  angles  of  30°  and  316°; 
tlnd  their  resultant  Ans.  B  =  1,69  P. 

2* 


ipiWsaa'Wiywwi 


34 


CONCtmRIlfO    FOROSa. 


i. 


3.  Given  three  concurring  forces  ol  4,  6,  and  6  lbs., 
whose  directions  are  inclined  to  the  axis  of  «  at  angles  of 
0°,  60°,  and  136°  respectively ;  find  their  resnltant. 

An$.  R=  V97  +  16  V  6  -  39  V2. 

4.  Given  three  equal  concurring  forces,  P,  whose  direc- 
tions are  inclined  to  the  axis  of  x  at  angles  of  30°,  60°,  and 


165°  ;  find  their  resultant. 


Ans.  i2  =  1.67  P. 


5,  Given  three  concurring  forces,  100,  50,  and  200  lbs., 
whose  directions  are  inclined  to  the  axis  of  x  at  angles  of 
0°  60°  and  180°;  find  the  magnitude  and  direction  of 
their  resultant  Ans.  R  =  86.6  lbs. ;  0  =  150°. 

38.  To  find  the  Magnitnde  and  Direction  of  the 
Resultant  of  any  number  of  Concnrrlng  Forces  in 
Bpace.-Let  P^,  P„  P„  etc.,  be  the  forces,  and  the 
whole  be  referred  to  a  system  of  rectangular  co-ordmates. 
Let  a„  /3t,  yj,  be  the  angles  which  the  direction  of  Pj 
makes  with  three  rectangular  axes  drawn  through  the  point 
of  appUcation  ;  let  «„  a„  r„  be  the  angles  which  the  direc- 
tion of  P,  makes  with  the  same  axes;  aj,  Pi,  Ts.  tne 
angles  which  P,  makes  with  the  same  axes,  etc.  Resolve 
these  forces  along  the  co-ordinate  axes  (Art.  34) ;  the  com- 
ponents of  P,  along  the  axes  are  Pj  cos  o„  Pj  cos  0,,  P, 
cos  y  .  Resolve  each  of  the  other  forces  in  the  same  way, 
and  let  X,  Y,  Z,  be  the  algebraic  snms  of  the  components 
of  the  forces  along  the  axes  of  x,  y,  and  z,  respectively ; 
then  we  have 

X  =  P,  cos  «i  +  P,  cos  «,  +  Pj  cos  «,  -H  etc.) 

=  SP  COS  a. 
Y=P.  cos  /3j  +  P,  COS  /3,  +  Pj  COS  0,  +  etc.(  ^^^ 

=  IP  cos  |3. 
Z  =  Pi  COS  yi  +  Pg  cos  y,  +  P»  cos  y,  +  etc.] 

=  SP  cos  y. 


6,  and  6  Iba, 
jf  x  at  angles  of 
Bgultant. 

V  6  —  39  V2. 

P,  whose  direc- 
of  30°,  60°,  and 
B  =  1.C7  P. 

50,  and  200  lbs., 
f  a;  at  angles  of 
md  direction  of 
M.;e  =  150°. 

Irection  of  the 
ring  Forces  in 

forces,  and  the 
liar  co-ordinates. 

direction  of  Pi 
through  the  point 
s  which  the  direc- 

Kj,  Ps,  Ts.  *!»« 
ses,  etc.  Resolve 
rt.  34)  ;  the  com- 

!„  Pi  COS0,,  P, 

in  the  same  way, 
!  the  components 
a  z,  respectively ; 


BOS  «j  +  etc.) 
cos  0,  +  etc.^  ^» 
cosy,  +  etc.] 


CONDITIOira  OF  EQlflUBRIUJl. 


3S 


Let  i2  be  the  resultant  of  all  the  forces;  and  let  the 
angles  which  its  direction  makes  with  the  three  axes  be  a, 
b,  c ;  then  as  the  resolved  parts  of  B  along  the  three  co-or- 
dinate axes  are  equal  to  the  sum  of  the  resolved  parts  of 
the  several  components  along  the  same  axes,  we  have 

^  cos  a  =  X,    iJ  COB  ft  =  r,    i2  cos  c  --  Z.       (2) 

Squaring,  and  adding,  we  get 

B?  =z  X^  +  m  -^  Z*', 


X  Y 

cos  fl  =  -p,    cos  ft  =  -^, 


cos  c  =  -p; 


(3) 
('4 


which  determines  the  magnitude  of  the  resultant  of  any 
system  of  forces  in  space  and  the  angles  its  direction  makes 
with  three  rectangular  axes. 

39.  The  Conditions  of  Equilibrium  for  any  num- 
ber of  Concnrring  Forces  in  Space.— If  the  forces  are 
in  equilibrium,  ^  =  0  ;  therefore  (3)  of  Art.  38  becomes 

2'»  +  r«+z»  =  o. 

But  as  every  square  is  essentially  positive,  this  cannot  be 
unless  X  =  0,  F  =  0,  Z  =  0 ;  and  therefore 

2:Pco8a  =  0,    IP  008/3  =  0,    SPco8y  =  0;    (1) 

and  these  are  the  conditions  among  the  forces  that  they 
may  be  in  equilibrium  ;  that  is,  the  sum  of  the  components 
of  the  forces  along  each  of  the  three  co-ordinate  axes  is 
equal  to  zero. 

40.  Tension  of  a  String. — By  the  iension  of  a  string 
is  meant  the  pull  along  its  fibres  which,  at  any  point,  tends 
to  stretch  or  break  the  string.  In  the  application  of  the 
preceding  principles  the  string  or  cord  is  often  used  as  a 


■  'swviStWMB^.'flKs-wtwsair!'.' '  ■ 


36 


BXAMPLKa. 


moiuis  of  oommnnioating  force.  A  string  is  said  to  be  per- 
fectly flexible  when  any  force,  however  small,  which  is 
applied  otherwise  than  along  the  direction  of  the  string, 
will  change  its  form.  In  this  work  the  string  will  be 
regarded  as  perfectly  flexible,  inextensible,  and  withont 
weight 

If  such  a  string  be  kept  in  eqnilibrinm  by  two  forces, 
one  at  each  end,  it  is  clear  that  these  forces  must  be  equal 
and  act  in  opposite  directions,  so  that  the  string  assumes 
the  form  of  a  straight  line  in  the  direction  of  the  forces. 
In  this  case  the  tension  of  the  string  is  the  same  throngh* 
out,  and  is  measured  by  the  force  applied  at  one  end  ;  and 
if  it  passes  over  a  smooth  peg,  or  over  any  number  of 
smooth  surfaces,  its  tension  is  the  same  at  all  of  its  points. 
If  the  string  should  be  knotted  at  any  of  its  points  to  other 
strings,  we  must  regard  its  continuity  as  broken,  and  the 
tension,  in  this  case,  will  not  be  the  same  in  the  two  por- 
tions which  stext  from  the  knot 


BXAMPLES. 

1.  A  and^  B  (Fig.  8)  are  two  fixed 
points  in  a  horizontal  line ;  at  A  is 
fastened  a  string  of  length  h,  with  a 
smooth  ring  at  its  other  extremity,  0, 
through  which  passes  another  string  with 
one  end  fastened  at  B,  the  other  end  of 
which  is  attached  to  a  given  weight  W ; 
determine  the  position  of  C. 

Before  setting  about  the  solution  of  statical  problems  of 
this  kind,  the  student  will  clear  the  ground  before  him,  and 
greatly  simplify  his  labor  by  asking  himself  the  following 
questions :  (1)  What  lines  are  tbiue  in  the  flgore  whose 
lengths  are  already  given  P  (2)  What  forces  are  there 
whose  magnitudes  are  already  given,  and  what  are  the 
forces  whose  magnitudes  are  yet  unknown?    (3)  What 


it  is  required  to 


mid  to  be  per- 
mall,  which  ia 
I  of  the  string, 
string  will  be 
,  and  without 

by  two  forces, 
must  be  equal 
string  assumes 

of  the  forces. 

same  through* 

one  end ;  and 
my  number  of 
1  of  its  points, 
points  to  other 
roken,  and  the 
in  the  two  por- 


is  required  to 

al  problems  of 
i)efore  him,  and 

the  following 
le  figure  whose 
irces  are  there 

what  are  the 
nP    (3)  What 


mXAMPLKS. 


87 


variable  lines  or  angles  in  the  figore  would,  if  they  were 
known,  determine  the  required  position  of  0  P 

Now  in  this  problem,  (1)  the  linear  magnitudes  which 
are  given  are  the  lines  AB  and  AC.  (2)  The  forces  acting 
at  the  point  C  to  keep  it  at  rest  are  the  weight  W,  a  ten- 
sion in  the  string  CB,  and  another  tension  in  the  string 
CA.  Of  these  W  is  given,  and  so  is  the  tension  in 
OB,  which  must  also  be  equal  to  W,  since  the  ring  is 
smooth  and  the  tension  therefore  of  WCB  is  the  sanio 
throughout  and  of  course  equal  to  W.  But  as  yet  there  is 
nothing  determined  about  the  magnitude  of  the  tension  in 
CA.  And  (3)  the  angle  of  inclination  of  tbo  string  OA  tc 
the  horizon  would,  if  known,  at  once  determine  the  posi- 
tion of  0.  For  if  this  angle  is  known,  we  can  draw  AC  of 
the  given  length ;  then  joining  0  to  B,  the  position  of  the 
system  is  completrly  known. 

Let  AB  =  fl,  AC  =  b,  CAB  =  e,  CBA  =  ^,  and  the 
tension  of  the  string  AC  =  T.  Then,  for  the  equilibrium 
of  the  point  C  under  the  action  of  the  three  forces,  W,  W, 
and  T,  we  apply  (2)  of  Art.  37,  and  resolve  the  forces 
horizontally  and  vertically  j  and  equate  those  acting  towards 
the  right-hand  to  those  acting  towards  the  left ;  and  those 
acting  upwards  to  those  acting  downwards.  Then  the 
horizontal  and  vertical  forces  are  respectively 


Trsin0  +  TsinS  =  W. 

Eliminating  T  we  have 

cos  d  =  sin  (9  +  0) ; 

.-.    25  +  0  =  90°. 

(1) 

Also,  from  trigonometry  we  have 

nn  (0  -H  0)  _  a . 
sin  </>           d ' 

(2) 

r 


38 


EXAMPLES. 


T\%M 


from  (1)  and  (3)  (?  and  0  may  be  fonn-^ ;  and  therefore  T 
may  be  found;  and  thus  all  the  circumstances  of  the 
problem  are  determined. 

2.  One  end  of  a  string  is  attached  to 
a  fixed  point,  A,  (Fig.  9) ;  the  string,  after 
passing  o?er  a  smooth  peg;  B,  sustains  a 
given  weight,  P,  at  its  other  extremity, 
and  to  a  given  point,  C,  in  the  string  is 
knotted  a  given  weight,  W.  Find  the  posi- 
tion of  equilibrium. 

The  entire  length  of  the  string,  ACBP,  is  of  no  conse- 
quence, since  it  is  clear  that,  once  equilibrium  is  estab- 
lished, P  might  be  suspended  from  a  point  at  any  distance 
whatever  from  B.  The  forces  acting  at  the  point,  0,  are 
the  given  weight,  W,  the  tension  in  the  string,  CB,  which, 
since  the  peg  is  smooth,  is  /*,  and  the  tension  in  the  string 
OA,  which  is  unknown. 

Let  AB  =  a,  AC  =  4,  CAB  =  »,  CBA  =  ^,  and  the 
tension  of  the  string,  AC  =  T.  Then  for  the  equilibrium 
of  the  point  Cj  we  have  (Art.  32), 


(1) 


P 

cosfl 

>r~ 

sin  (e  -1-  0) ' 

also, 

from  the  geometry  of  the  figure,  we  ] 

ft  sin  (0  +  0)  =  a  sin  0. 

From(l) 

and  (2)  we  get 

P 

dcc«(  Q 

W~ 

a  sin^' 

or 

rin^  = 

hW        . 
-p  cos  tf; 

(«) 


COS0  = 


Vo»/'»-ft»Fr»  coB«  9 

___ 


\ 


c      I    II  ilif  IjjIWBW 


id  therefore  T 
Btances  of  the 


ve 


P* 


■■*>*^p* 


■Hfii 


is  of  no  conse- 
briam  is  estab- 
at  any  distance 
be  point,  C,  ore 
Qg,  CB,  which, 
n  in  the  string 

.  =  ip,  and  the      ^ 
the  equilibrium 


(1) 


(3) 


>B»» 


MXAMPLIJS,  99 

Expanding  sin  (fl  +  ^)  in  (2),  and  substitating  in  it  these 
values  of  sin  ^  and  cos  0,  and  reducing,  we  have  the 
equation 

^   =0, 


^,,S!!f^^^±D^», 


2W*b 


from  which  6  may  be  found.      (See  Minchin's  Statics, 
p.  29.) 

3.  If,  in  the  last  example,  the  weight,  W,  instead  of 
being  knotted  to  the  string  at  C,  is  suspended  from  a 
smooth  ring  which  is  at  liberty  to  slide  along  the  string, 
AOB,  find  the  position  of  equilibrium. 

Atu.  sin  0  =  ^. 

41.  BqniUbrinm  of  Coneiurrliig  Forces  on  a 
Smooth  Piano. — If  a  particle  be  kept  at  rest  on  a  smooth 
surface,  plane  or  curved,  by  the  action  of  any  number  of 
forces  applied  to  it,  the  resultant  of  these  forces  must  be  in 
the  direction  of  the  normal  to  the  surface  at  the  point 
where  the  particle  is  situated,  and  must  be  equivalent  to 
the  pressure  which  the  surface  sustains.  For,  if  the 
resultant  had  any  other  direction  it  could  be  resolved  into 
two  components,  one  in  the  direction  of  the  normal  and  the 
other  in  the  direction  of  a  tangent ;  the  first  of  these  would 
be  opposed  by  the  reaction  of  the  surface ;  the  second  being 
unopposed,  would  cause  the  particle  to  move.  Hence,  we 
may  dispense  with  the  t»lane  altogether,  and  regard  its 
normal  reaction  as  one  of  the  forces  by  which  the  particle 
is  kept  at  rest.  Therefore  if  the  particle  on  which  the 
statical  forces  act  b«  on  a  smooth  plane  surface,  the  case  is 
the  same  aa  that  treated  in  Art.  39,  viz.,  equilibrium  of  a 
particle  acted  upon  by  any  number  of  forces ;  and  in  vnnt- 
ing  down  the  equations  of  equilibrium,  wo  merely  have  to 
include  the  normal  i  action  of  the  plane  among  all  the 
others. 


p^ 


40 


JtXAMPLSa. 


BXAMPLBS. 

1.  A  heavy  particle  is  placed  on  a 
smooth  inclined  plan»,  AB,  (Pig.  10), 
and  is  sustained  by  a  force,  P,  whidi 
acts  along  AB  in  the  vortical  plane 
which  is  at  right  angles  to  AB ;  find 
P,  and  also  the  pressure  on  the  in- 
clined plane. 

The  only  eflfect  of  the  inclined  plane  is  to  produce  a 
normal  reaction,  R,  on  the  particle.  Hence  if  we  intro- 
duce this  force,  we  may  imagine  the  plane  removed. 

Let  IT  be  the  weight  of  the  particle,  and  «  the  inclina- 
tion of  the  plane  to  the  horizon. 

Resolving  the  forces  along,  and  perpendicular  to  AB, 
since  the  lines  along  which  forces  may  be  resolved  are 
arbitrary  (Art  37),  we  have  guooeflsively, 

P— ITsittft  SE  0,    or    PzszWfAna', 


and 


R  —  W'cos  «  =  0,    or    R  =  IT  cos  a. 


If,  for  example,  the  weight  of  the  particle  is  4  or,  and 
the  in  hnation  of  the  plane  30°,  there  will  be  a  normal 
pressure  of  2^3  oz.  on  the  plane,  and  the  force,  P,  wUl 
be  2  oz. 


a.  In  the  previous  example,  if  P  act  horizontally,  find 
its  magnitude,  and  also  that  of  R. 

Resolving  along  AB  and  perpendicular  to  it,  we  have 
Bucccsdiveiy, 


P  cos  a— IT  sin  a  =  0,    or    PssTTtana; 


\^ 


and     PBin«+ F  oo««  — i?  =  0,    .•.    J?  = 


W 
oobm* 


8  to  produce  a 
ce  if  we  intro> 
amoved. 
a  the  inclina- 

licnlar  to  AB, 
>e  resolved  are 


)OS  a. 

le  ia  4  oz.,  and 
I  be  a  normal 
>  force,  P,  will 

risontally,  find 
o  it,  we  have 


Ftan  «; 


V^ 


coimmoifa  op  gqpuMnnm. 


U 


3.  If  the  particle  is  auatained  by  a  ft)roe,  P,  mining  a 
given  angle,  6,  with  the  inclined  plane,  find  the  mi^itade 
of  this  force,  and  of  the  pressure  on  the  plane,  ail  the  forces 
acting  in  the  same  vertical  plane. 

Besolving  along  and  perpendicular  to  the  jdaae  succes- 
sively, we  have 

PcobO—  Wnna  =  0, 
and  JJ  +  P  sine— ff  COB  «  =  0, 


from  which  we  obtain 

P^W^^,;    R 

008  $' 


_-  |^rCoe(«  +  ^) 


COS0 


Rem.— The  advantage  of  a  judicious  selection  of  direc- 
tions for  the  resolution  of  the  forces  is  evident.  By  resolv- 
ing at  right  angles  to  one  of  the  unknown  forces,  we 
obtain  an  equation  free  fh)m  that  force;  whereas  if  the 
directions  are  selected  at  random,  all  of  the  forces  will 
enter  each  equation,  which  will  make  the  solution  less 
simple. 

The  student  will  observe  that  these  values  of  P  and  B 
cculd  have  been  obtained  at  once,  without  resolution,  by 
Art.  32. 

42.  Conditiosui  oi  SqidUbiiaiii  for  may  number  of 
Ooncnrring  ForoM  whoa  the  perticle  on  which  they 
act  ia  Oeaatrained  to  Remain  on  a  Qiven  Smooth 
Bvrfiwe.— If  a  particle  be  kept  at  rest  on  a  smooth  sur- 
face by  the  action  of  any  number  of  forces  applied  to  it, 
the  resultant  of  these  forces  must  be  in  the  direction  of  the 
normal  to  the  surface  at  the  point  where  the  particle  is 
situated,  and  must  be  equivalent  to  the  pressure  which  the 
surface  sustains  (Art.  40).  Hence  since  the  resultant  is  in 
the  direction  of  the  normal,  and  is  destroyed  by  the  roao- 


iSt 


CONDinOtfS  OF  MqmUBRIUM. 


tion  of  the  Brrface,  we  may  regard  this  reaction  as  an 
addildonal  force  directly  opposed  to  the  normal  force. 

Let  N  be  the  normal  reaction  of  the  surface,  and  a,  0,  y, 
the  aiigles  which  JV  makes  with  the  co-ordinate  axes  of  x, 
y,  and  t,  respectively.  Let  X,  Y,  Z,  be  the  sum  of  the 
comfoneuts  of  all  the  other  forces  resolved  parallel  to  the 
three  axes  respectively.  The  reaction  JVmay  i.,  considered 
a  ne^r  force,  which,  with  the  other  forces,  keeps  the  parti- 
cle im  eqoilibriom.  Therefore,  resolving  N  parallel  to  the 
three  axes,  we  have  (Art  39), 


X+  JVcosa  =  0, 
F-f-JVcos/S 
Z+  JVcosy 


=  0.  > 


(1) 


Leit  u  =/(«,  y,  «)  =  0,  be  the  equation  of  the  given 
Burfa<3e,  and  x,  y,  t  the  co-ordinates  of  the  particle  to 
which  the  forces  are  applied.    We  have  (Anal.  Geom., 

Art.  :176), 

a' 
cos  a  = 


oosjS  = 


cosy  = 


y 

Vo'«  -I-  »'» +  I'i 

1 


(2) 


where  a'  and  V  are  the  tangents  of  the  angles  which  the 
projections  of  the  normal,  N,  on  the  co-ordinate  planes  xz 
and  yt  make  with  the  axis  of  tt.  Since  the  normal  is  per- 
pendicular to  the  plane  tangent  to  the  surface  at  (x,  y,  t), 
t))o  projections  of  the  normal  are  perpendicular  to  the 
traces  of  the  plane.     Therefore  (AnaL  Qeom.,  Art   37, 


Cor.  1),  W6  have 


and 


1  -I-  aa'  =  0, 
1  +  W  =  0 ; 


(8) 
(4) 


mm/Hmfm^ 


UM. 

s  reaction  as  an 
rmal  force. 
Pace,  and  a,  0,  y, 
dinate  axes  of  z, 

the  sum  of  the 
ed  parallel  to  the 
nay  <.,  considered 

keeps  the  parti- 
¥  parallel  to  the 


(1) 


ion  of  the  given 
[  the  particle  to 
e  (Anal.  Geom., 


(2) 


angles  which  the 
rdinate  planes  xz 
he  normal  is  per- 
irfaco  at  {x,  y,  *), 
endicular  to  the 
Qeom.,  Art.   37, 

(8) 
(4) 


in  Thich 
a 


CONDrnOIfB  OF  EqUIUBRIUK. 

I  '  y 
dx       ,      daf     ,_dy     j.__^ 


(Calculus,  Art.  66a.)    Substituting  in  (3)  and  (4),  we  have 


x.S-^  =  o; 

and 

>-^|-f  =  <". 

from  which 

du 

d^ 

-§^=  ?  (OaLArt87)  =  a', 

dx       du  ^ 

dt 

(5) 


du 

,„a      ^  =  -^  =  #  =  *'.  <«) 

M  dy       du 

dM 

Substituting  these  values  of  a'  and  7/  in  (2)  and  multiply- 

du       , 
ing  both  terms  of  the  fraction  by  ^,  we  have 


008a  = 


008/3  = 


dm 
tSi 


3. 


cosy  = 


(7) 


I 


f 


M 


44 


CONDITTOm   OP  SQVIMBMIXrM. 


\ 


which  give  the  value  of  the  direction  cosines  of  the  aonaai 
at  {x,  y,  z). 

Putting  the  denominator  equal  to  Q,  tot  shortness,  and 
fiubstitating  in  (1)  and  transposing,  we  have 


^  N    du 

r  —  ——    ~ 
Q'  dy' 

Z  —  ——     ^ 
~       Q'  dz 


(8) 


Pi 


(10) 


X 

T        Z 

du 

du         du 

dm 

^      ST 

Eliminating  N  between  ihese  three  equations,  we  obtain 
the  two  independent  equations, 


m 


which  express  the  conditions  that  must  exist  among  the 
applied  forces  and  their  directions  in  order  tiiat  their 
resultant  may  be  normal  to  the  surface,  t.  e.,  that  there  may 
be  equilibrium.  If  these  two  equations  are  not  satisfied, 
equilibrium  on  the  surface  cannot  exist  Hence  the  point 
on  a  given  surface,  at  which  a  given  particle  under  the 
action  of  given  forces  will  rest  in  equilibrium,  is  the  point 
at  which  equations  (11)  are  satisfied. 

Cob.  1.— Squaring  equations  (8),  (9),  (10)  and  adding,  we 


get 


/rfM\>      /rftt\»      (du\ 

\dxf       ]dy}_      \dz} 

SW  "^     <?»    "^  "^J 


jv=  V  Jr»  +  r»  +  z*, 


IP; 


m 


■I.  wiwpiy  *»i  MfitnmMmKimtif 


^ 


r. 

'  shortness,  and 

(8) 
(9) 

(10) 
ions,  we  obtain 

(11) 


ist  among  the 
ler  tiiat  their 
that  there  may 
i  not  satisfied, 
mce  the  point 
icle  under  the 
u,  is  the  point 


and  adding,  ve 


»»   J 


(12) 


MXAMPLSa. 


W 


which  is  the  yalae  of  the  normal  reeistanoe  of  the  surface 
and  is  precisely  the  same  as  the  resultant  of  the  acting 
forces,  as  it  clearly  .ehoald  be  ;  but  this  resistance  must  act 
in  the  direction  opposite  to  that  of  i^e  resultant 

Cob.  2.— Multiplying  (8),  (9),  (10)  by  dx,  dy,  dz,  respeo- 
tirely,  and  adding,  and  ramembering  that  the  total  differ- 
ential of  «  =  0  is  zero,  we  get 


Zdx  +  Tdy  +  Zdx  =  0, 


(13) 


which  is  an  equation  of  condition  for  equilibrium.  If  (13) 
cannot  be  satisfied  at  any  point  of  the  surface,  equilibrium 
is  impossible. 

OoR.  3. — If  the  forces  all  act  in  one  plane,  the  surface 
becomes  a  plane  curve ;  let  this  curve  be  iu  the  plane  xy, 
then  z  =  0;  therefore  (11)  and  (13)  become 


X 

du 


7_ 

du' 


and 


Zdx  +  Ydy  =  0, 


(U) 


(15) 


in  which  (14)  or  (IS)  may  be  used  according  as  the  equation 
of  the  curve  is  given  as  an  implicit  or  explicit  function. 


EXAMPLES. 


1.  A  particle  is  placed  on  the  surface  of  an  ellipsoid,  and 
is  acted  on  by  attracting  forces  which  vuy  directly  as  the 
distance  of  the  particle  from  the  principal  pliraes*  of  cec- 
tion ;  it  is  required  to  determine  the  position  vf  equilibrium. 

Let  the  equation  of  the  ellipsoid  be 


«  = 


=  /rx,y.,;  =  ^;  +  g  +  ^-l:=0; 


•  nuM  of  cy,  yti  M). 


lipillWI  lliHjnBl|lu||l 


^'If^fT'^W 


\M 


I 


46  SXAMPLKS. 

•'•    dx~  a*'    dy~  V    dt  ~  (^* 

and  let  the  a;-,  y-,  and  c-components  of  the  foroei  be 
respectively, 

JT  =  —  UiXy  r  =  —  t«,y,  Z=  —Ut»\ 

then  (11)  will  give 

Uid^  =  «,{'  =  UfO*; 
which  may  bo  pnt  in  the  form 

»l^  _  «£  _  «|^  _  «i  +  «t  4- «» . 

If  these  conditions  are  fulfilled,  the  particle  will  rest  at  all 
points  of  the  sarface. 

3.  Again,  take  the  sam;)  surfince,  and  let  the  forces  vary 
inversely  as  the  distances  of  the  point  from  the  principal 
planes;  it  is  required  to  determine  the  position  of  eqnili- 
briom. 

Here    X=-^,    F=-^,    Z=-?^; 


therefore  (11)  becomes 


3fi 

u., 


«8 


»1  Mi  «8  «1  +  «t  +  **» 

by  putting  u  for  Ui  +  u,  +  tig, 


1 

It 


-  ^=«("#  »  =  »(^f.  '  =  'fe)*. 


2t 
J* 

the  foroee  be 


-«»*; 


!».. 

».-« 


nrill  rest  at  all 


the  forces  vary 
t  the  principal 
ition  of  eqaili- 


11' 
_  > 


1 


-m*- 


immm 


EXAMPLSa. 


47 


which  in  (12)  gives 


=^  + 


y* 


=  «[^-^^-^} 


8.  A  particle  ia  placed  ingide  a  smooth  spboipe  on  the  con- 
cave surface,  and  is  acted  on  by  gravity  and  by  a  repulsive 
force  which  varies  inversely  as  the  square  of  the  distance 
from  the  lowest  point  of  the  sphere;  find  the  position  of 
equilibrium  of  the  particle.  , 

Let  the  lowest  point  of  the  sphere  be  taken  for  the  origin 
of  co-ordinates,  and  let  the  axis  of  «  be  vertical,  and  posi- 
tive upwards;  then  the  equation  of  the  sphere,  whose 
radius  is  a,  is 

Lot  TT  =  the  weight  of  the  particle,  and  r  =  the  distance 
of  it  from  the  lowest  point;  then 

r«  =  a^  +  ^  +  «'  =  2o«. 

Also,  let  the  -epnlsive  force  at  the  unit's  distance  =  « ; 
then  at  the  distuioe  r  it  will  be 


—  ^ 
X  = 

r  = 


Sox' 


2as 
u 

w 


-If. 


ji. 


i 


mmmmm^immsBss 


48 


MXAMPLXa. 


Let  iV  =  the  normal  pressni-e  of  the  curve ;  then  (8)  and 
(10)  give 


i.t-^+^'-^-o.. 


from  which  we  have 


t  = 


^a^W^' 


whence  the  position  of  the  particle  is  known  for  a  given 
weight,  and  for  a  given  value  of  u.  (See  Price's  Anal. 
Mechanics,  Vol.  I,  p.  39.) 

4.  Two  weights,  jP  and  C,  are  fastened  to  the  ends  of  a 
string,  (Fig.  11),  which  passes  over  a  pulley,  0;  and  ^ 
hangs  Ireely  when  P  rests  on  a  plane  curve,  ^P,  in  a 
vertical  plane  ;  it  is  required  to  find  the  position  of  equili- 
brium when  the  curve  is  given. 

The  forces  which  act  on  P  are  (1)  the 
tensiou  of  the  string  in  the  line  OP,  which 
is  equal  to  the  weight  of  Q,  (2)  the  weight 
of  P  acting  vertically  downwards,  (3)  the 
normal  reaction  of  the  curve  R. 

Let  0  be  the  origin  of  oo-ordinates,  and 
the  axis  of  x  vertical  and  positive  down- 
wards. Let  OM  —  X,  MP  =  If,  OP  =  r, 
POM  =e,OA=a.    Then, 


Flg.ll 


X=  P-Qfmd—R 


dy 


T^-Qmxd+R^', 


then  <8)  and 


n  for  a  given 
Price's  Anal. 


the  ends  of  a 
jr,  0;  and  Q 
ve,  AP,  in  a 
ion  of  equili- 


Fig.ll 


MXAMPm.9, 

therefore  from  (15)  we  have 

(P  —  Q  COB  6)  dx  —  Q  sin  ddy  =  0, 


4« 


or 

But  since 
we  have 


a*  +  »•  =  f4, 
zdx  -\-  ydy  =.  rdr\ 
.'.    Pdz—Qdr  =  0; 


(1) 


which  is  the  oonditidn  that  most  be  aatisfed  by  P,  Q,  and 
the  equation  of  the  curve. 

6.  Required  the  equation  of  the  curve,  on  ^sill  points  of 
which  P  will  rest. 

Integrating  (1)  of  Ex.  4,  we  have 

Px-^Qr=  O.  (1) 

But  since  P  is  to  rest  at  all  points  of  the  curve,  this  equr 
tion  must  be  satisfied  when  P  is  at  A,  from  which  we  get 
a;  =  r  =  o ;  therefore  (1)  becomes 


Pa—Qa=z  C; 


which  in  (1)  gives 


P         ' 
1  —  ^  cos  0 

which  is  the  equation  of  a  conic  section,  of  which  the  focus 
is  at  the  pole  0 ;  and  is  an  ellipse,  parabola,  or  hyperbola, 
according  ac  P  <,  =,  or  >  Q. 
3 


IK> 


SXAMPLBa. 


EXAMPLES. 

1.  Two  forces  of  10  and  20  lbs.  act  on  a  particle  at  an 
angle  of  60° ;  find  the  resuliont.  Ans.  26.5  lbs. 

2.  The  resultant  of  two  forces  is  10  lbs.;  one  of  the 
forces  is  8  lbs.,  and  the  other  is  inclined  to  the  resultant  at 
an  angle  of  36°.  Find  it,  and  also  find  the  angle  between 
the  two  forces.  (There  are  two  solutions,  this  being  the 
ambiguous  case  in  the  solution  of  a  triangle.) 

Ans.  Force  is  2.66  lbs.,  or  13.52  lbs.  Angle  is  47°  17' 
05",  or  132°  42' 55". 

3.  A  point  is  kept  at  rest  by  forces  of  6,  8,  11  lbs. 
Find  the  angle  between  the  forces  6  and  8. 

Ana.  77°  21' 52". 

4.  The  directions  of  "iwc  forces  acting  at  a  point  are 
inclined  to  each  other  (1)  at  an  angle  of  60°,  (2)  at  an 
angle  of^  120°,  and  the  respective  resultants  are  as 
V?  :  VS  ;  compare  the  magnitude  of  the  forces. 

Ans.  2  :  1. 

5.  Three  posts  are  placed  in  the  ground  so  as  to  form  an 
equilateral  triangle,  and  an  elastic  string  is  stretched  round 
them,  the  tension  of  which  is  6  lbs. ;  find  the  pressure  on 
each  post.  ^„«.  e  Vs. 

6.  The  angle  between  two  unknown  forces  is  37°,  and 
their  resultant  divides  this  angle  into  31°  and  6°  ;  find  the 
ratio  of  the  component  forces.  Ans.  4927  :  1. 

7.  If  two  equal  rafters  support  a  weight,  W,  at  their 
upper  ends,  required  the  compression  on  each.  Let  the 
length  of  each  lufter  be  a,  and  the  horizontal  distance 
between  their  lower  ends  be  &  ,  aW 


Ans. 


V4o»-i» 


particle  at  an 
ns.  26.5  lbs. 


I. 


10 


one  of  the 
.  reaaltaut  at 
angle  between 
this  being  the 


ingle  is  47°  17' 


'  6,  8,  11  Iba. 

77°  21'  52". 

at  a  point  are 

60°,  (2)  at  an 

Itants   are   as 

rces. 

Ana.  2  :  1. 

>  as  to  form  an 
TetcLed  round 
le  pressure  on 
Ana.  6  \/3. 

ses  is  37°,  and 
i  6° ;  find  the 
».  4.927  :  1. 

;,   W,  at  their 
och.     Let  the 
ontal  distance 
aW 

V4o»  -  V 


■m 


■■• 


MXAMPUa^ 


wmmm 


51 


8.  Three  forces  act  at  a  point,  and  include  angles  of 
90°  and  45°.  The  first  two  forces  are  each  equal  to  2P, 
and  the  resultant  of  them  all  is  VlOP;  find  the  third 
fofce.  ^ng.  P  V%. 

9.  Find  the  magnitude,  R,  and  direction,  0,  of  the 
resultant  of  the  three  forces,  P,  =  30  lbs.,  P,  =  70  lbs., 
Pj  =  50  lbs.,  the  angle  included  between  P,  and  P, 
being  56°,  and  between  P,  and  P,  104°.  (It  is  generally 
conTenient  to  take  the  action  line  of  one  of  the  forces  for 
the  axis  of  a;.) 

Let  the  axis  of  x  coincide  with  the  direction  of  P^ ;  then 
(Art.  36),  we  have 

X  =  22.16 ;    Y  =  75.13 ;    B  =  78.33 ;    «  =  73°  34'. 

10.  Three  forces  of  10  lbs.  each  act  at  the  same  point ; 
the  second  makes  an  angle  of  30°  with  the  first,  and  the 
third  makes  an  angle  of  60°  with  the  second ;  find  the 
magnitude  of  the  resultant.  Ana.  24  lbs.,  nearly. 

11.  If  three  forces  of  99,  100,  and  101  units  respectively, 
act  on  a  point  at  angles  of  120°;  find  the  magnitude  of 
their  resultant,  and  its  inclination  to  the  force  of  100. 

Ana.  VS;  90°. 

12.  A  block  of  800  lbs.  is  so  situated  that  it  receives 
from  the  water  a  pressure  of  400  lbs.  in  a  south  direction, 
and  a  pressure  from  the  wind  of  100  lbs.  in  a  westerh 
direction ;  required  the  magnitude  of  the  resultant  pres- 
sure, and  its  direction  with  the  vertical. 

Ana.  900  lbs.;  27°  16'. 

13.  A  weight  of  40  lbs.  is  supported  by  two  strings,  one 
of  which  makes  an  angle  of  30°  with  the  vertical,  the  other 
45° ;  find  the  tension  in  each  string. 

.  Ana.  20  {VQ  -  Vi) ;  40  ( V3  —  1). 


tMM 


6% 


BXAMPLSa. 


14.  Two  forces,  P  and  P\  acting  along  the  diagonalfl  of 
a  parallelogram,  keep  it  at  rest  in  such  a  position  that  one 
of  its  sides  is  horizontal ;  show  that 

P  sec  «'  =  P'  sec  o  =  W  cosec  («  +  «')> 

where  W  is  the  weight  of  the  parallelogram,  and  a  and  «' 
the  angles  between  *;he  diagonals  and  the  horizontal  side. 

16.  Two  persons  pall  a  heavy  weight  by  ropes  inclined 
to  the  horizon  ut  angles  of  60°  and  30°  with  forces  »jf 
160  lbs.  and  200  lbs.  The  angle  between  the  two  veitical 
planes  of  the  ropes  is  30"  ;  find  the  single  horizontal  force 
that  would  produce  the  same  eflfect  Ans.  245.8  lbs. 

16.  In  order  to  raise  vertically  a  heavy  weight  by  means 
of  a  rope  passing  over  a  fixed  pulley,  three  workmen  pull  at 
the  end  of  the  rope  with  forces  -of  40  lbs.,  50  lb8.,8tul 
100  Ibr. ;  the  directions  of  these  forces  being  inclined  to 
the  horizon  at  «n  ax^gle  of  60".  What  is  the  magnitude  of 
the  resultant  force  which  tends  directly  to  riise  the  weight? 

Ans.  164.64  lbs. 

17.  Three  persons  pull  a  heavy  weight  by  cords  inclined 
to  the  horizon  at  an  angle  of  60°,  with  forces  of  100, 120, 
atid  140  lbs.  The  three  vertical  planes  of  the  cords  are 
inclined  to  each  other  at  angles  of  .S0°;  find  the  single 
horizontal  force  that  would  pi-oduoe  the  same  effect 

Am.  10  '^^146  +  72  Vl  lb«. 

18.  Two  forces,  P  and  Q,  acting  respectively  parallel  to 
the  base  and  length  of  an  inclined  plane,  will  each  singly 
sustain  on  it  a  particle  of  weight,  W\  to  det«rmino  the 
weight  of  W. 

Let  «  =  inclination  of  the  plane  to  the  horizon ;  then 
resolving  in  each  case  along  the  plane,  so  that  the  normal 
pressures  may  not  enter  into  the  equations  (See  Bern.,  Ex.  3, 
Art.  41),  wo  have 


T 


!e  diagonals  of 
sition  that  one 


1,  aod  a  and  «' 
triiiODtal  eide. 

'  ropes  inclined 
with  forces  of 
he  two  veitical 
horizontal  force 
nti.  245.8  lbs. 

eight  by  means 
workmen  pull  at 
>8.,  50  lbs.,  and 
iug  inclined  to 
e  mugnitude  of 
;J8e  the  weight  ? 
s.  164.64  lbs. 

r  cords  inclined 
XJ8  of  100,  120, 
f  the  cords  are 
find  the  single 
le  effect 


+  72  VS  lb«. 

vely  parallel  to 
rill  each  singly 
i  determine  tho 


B  horizon ;  then 
that  tlie  normal 
See  Bern.,  Ex.  3, 


MXAMPLBS. 


St 


Poofla  =  fTsina;     C=lfBina; 


jr  = 


PQ 


{p* 


W 


19.  A  cord  whose  length  is  21,  is  faster*  •  m  .■  vnd  B,  in 
the  same  horizontal  line,  at  a  distance  i  iii  .*cli  other 
oqual  to  2rt ;  and  a  smooth  ring  upon  the  cortt  sustains  a 
weight  W;  find  the  tension  of  the  cord. 

Ana.  T .:   --^^==. 
2  VP  —  fit' 

20.  A  heavy  particle,  whose  weight  is  W,  is  sustained  on 
ii  smooth  inclined  plane  by  three  forceg  applied  to  it,  each 

equal  to  ^;  one  acts  vertically  upward,  another  horizon- 

O 

tally,  and  the  third  along  the  plane ;  find  the  inclination, 


«,  of  the  plane. 


Ans.  tan 


1 
2' 


21.  A  body  whose  weight  ia  10  Ihs.  Is  supported  on  a 
smooth  inclined  plane  by  a  force  of  2  lbs.  acting  along  the 
plane,  and  a  horizontal  force  of  5  lbs.  Find  the  inclination 
of  the  plane.  ^»«.  sin-»  |. 

22.  A  body  is  sustained  on  a  smooth  inclined  plane  (in- 
clination a)  by  a  force,  P,  acting  along  the  plane,  and  a 
horizontal  torce,  Q.  When  the  inclination  is  halved,  and 
the  forces,  P  and  Q,  each  halved,  the  body  is  still  observed 
to  rest;  find  the  ratio  of  P  to  ^.        ^^^ 

23.  Two  weights,  P  and  Q,  (Fig,  12),  rest 
on  a  sipooth  doxihle-inolined  plane,  and  are 
attiiched  to  the  extremitios  of  a  string 
which  ijaases  over  a  smooth  peg,  0,  at  a 
point  vertically  over  the  intersection  of  the 

Hues,  the  jieg  and  the  weights  being  in  a 


Fi|.a 


■PBiWiP 


M 


MXAMPLSa, 


vertical  plane.    Find  the  position  of  equilibrium,  if  Z  =  the 
length  of  the  string  and  h  =  CO. 

Ans.  The  position  of  equilibrium  is  given  by  the  equa- 
tions 

p  8in_a  _  -,  sin  /3  ► 

cos  *  ~  ^  cos  ^  * 

cos  «      oos_^  _  I 
sin  0      sin  0  ~  ^* 

24.  Two  weights,  P  and  Q,  connected  by  a  string, 
length  /,  rest  on  the  convex  side  of  a  smooth  vertical 
circle,  radius  a.  Find  the  position  of  equilibrium,  and 
show  that  the  heavier  weight  will  be  higher  up  on  the 
cirele  than  the  lighter,  the  radius  of  the  circle  drawn  to  P 
making  an  angle  0  with  the  vertical  diameter. 


Ana.  P  sin  6  =  Q  sin  ( eh 


25.  Two  weights,  P  and  Q,  connected  directly  by  a 
string  of  given  length,  rest  on  the  convex  side  of  a  smooth 
vertical  circle,  the  string  forming  a  chord  of  tii«?  circle ; 
find  the  position  of  equilibrium. 

Ans.  If  2a  is  the  angle  subtended  at  the  centre  of  the 
circle  by  the  string,  the  inclination  0,  of  the  string  to  the 
vortical  is  given  by  the  equation 

P     0 
cot  fl  =  p-  ^^  tan  o. 


26.  Two  weights,  P  and  Q,  (Fi^.  13), 
Teat  on  the  concave  Hide  of  a  pari<  )la 
whose  axis  is  horizontal,  and  are  con- 
nected by  a  string,  length  I,  which 
]iasse8  over  a  smooth  peg  at  the  focus,  F. 
Find  the  {wsition  of  equilibrium. 

A»8.  Let  0  —  the  angle   which   FP 


Fi|.,ll 


turn,  if  2  r=  the 
n  by  the  equa- 


by  a  string, 
nooth  vertical 
lilibriuni,  and 
ler  up  on  the 
le  drawn  to  P 


wmm- 


»n 


G'-»)- 


directly  by  a 
le  of  a  smooth 
of  tii'5  circle ; 

centre  of  the 
3  string  to  the 


'IfcU 


BXAMPLSa. 


65 


makes  with  the  aria,  and  4m  =  the  latus  reotnm  of  the 

parabola,  then  

e         Py/l-  2m_ 

27.  A  particle  is  placed  on  the  convex  side  of  a  smooth 
ellipse,  and  is  acted  upon  by  two  forces,  F  and  F',  towards 
the  foci,  and  a  force,  F"y  towards  the  centre.  Find  the 
position  of  equilibrium. 

^^   ,.  _  __A__,  where  r  =  the  distance  of  the  par- 

Vl  —  «• 

tide  ftom  the  centre  of  the  eUipse  ;  b  =  semi-minor  was, 

F-F' 
and  n  =  —^ — 

28.  Let  the  curve,  (Fig.  11),  be  a  circle  in  which  the 
origin  and  pulley  are  at  a  distance,  c,  above  the  centre  of 
the  circle :  to  determine  the  position  of  equilibrium. 

Q 
Ans.  r  =  -pa. 

29.  Let  the  curve,  (Fig.  11),  be  a  hyperbola  in  which  the 
origin  and  pulley  are  at  the  centre,  0,  the  transverse  axis 
being  vertical ;  to  determine  the  position  of  equilibrium. 

30.  A  particle,  P.  is  acted  upoL  oy  two  forces  towards 

two  fixed  points,  8  and  H,  these  forces  being  ^  and  ^p 

respectively ;  prove  that  P  will  rest  at  all  points  inside  a 
smooth  tube  in  the  form  of  a  curve  whose  equation  is  8P. 
PH  =  ]^,k  being  a  constant 

31.  Two  weights,  P  and  Q,  connected  by  a  string,  r«8t 
on  the  convex  side  of  a  smooth  cycloid.  Find  the  position 
of  equilibrium. 


56 


aXAMPUUL 


Am.  If  2  s  the  length  of  the  string,  and  «  s  radios  of 
generating  circle,  the  position  of  eqoilibriam  is  defined  by 
the  equation 

.0         Q        I 
""  a  =  pTq  '  45' 

where  B  is  the  angle  between  the  vertioal  and  the  radius  to 
the  point  on  the  generating  circle  which  corresponds  to  P. 

32.  Two  weights,  P  and  Q,  rest  on  the  convex  side  of  a 
smooth  vertical  circle,  and  are  connected  by  a  string  which 
passes  over  a  smooth  peg  vertically  over  the  centre  of  the 
circle  ;  find  the  position  of  equilibrinm. 

Ana.  Let  h  =.  the  distance  between  the  peg,  B,  and  the 
centre  of  the  circle  ;  0  and  ^  =  the  angles  made  with  the 
vertical  by  the  radii  to  P  and  Q,  respectively  ;  a  and  j3  = 
the  angles  made  with  the  tangents  to  the  circle  at  P  and 
Q  by  the  portions  PB  and  QB  of  the  string ;  /  =b  length 
of  the  string;  then 


COS  a  ' 

_  nsin^ 

-   ^C08/3' 

\cos  a 

Bin  *\ 
cos  /)/  -  *» 

ft  cos  (fl  -f 

a)  =.  a  cos  a. 

A  COS  (^  +  /3)  =  a  cos  j3. 

«  =:  radios  of 
u  defined  by 


.  the  radias  to 
esponds  to  P. 

nvex  ride  of  a 

a  sti-ing  which 

centre  of  the 

Bg,  B,  and  the 
made  with  the 
;  a  and  j3  = 
irole  at  P  and 
b; ;  2  =B  length 


CHAPTER    III. 

COMPOSITION   AND    RESOLUTION    OF  FORCES  ACTING 
ON    A    RIGID    BODY. 

43.  A  Xtigid  Body. — In  the  last  chapter  wo  considered 
the  action  of  forces  which  have  a  common  point  of  applica- 
tion. We  shall  now  consider  the  action  of  forces  which  are 
applied  at  different  points  of  a  rigid  body. 

A  rigid  body  is  one  in  which  the  particles  retain  invari- 
able positions  with  respect  to  one  anotbeis  so  that  no 
pxternal  force  can  alter  them.  Now,  as  a  matter  of  fact, 
there  is  no  such  thing  in  nature  as  a  body  that  is  perfectly 
rigid ;  every  body  yields  more  or  less  to  the  forces  which 
act  on  it.  If,  then,  in  any  case,  the  body  is  altered  or  com- 
pressed appreciably,  we  shall  suppose  tiuit  it  has  assumed 
its  figure  of  equilibrium,  and  then  consider  the  points  of 
application  of  the  forces  as  a  system  of  invariable  form. 
The  term  body  in  this  work  means  rigid  body. 

44.  TransmisBibility  of  Force.— When  &  force  acts 
at  a  definite  point  of  a  body  and  along  a  definite  line,  the 
iffect  of  the  force  will  be  unchanged  at  whatever  point  of 
its  direction  we  suppose  it  applied,  ])rovided  this  point  be 
I'ither  one  of  the  points  of  the  body,  or  be  invariably  con- 
iioctcd  with  the  body.  This  principle  is  called  the  trans- 
missibilily  of  a  force,  to  any  point  in  its  line  of  action. 

Now  two  Ciiual  forces  acting  on  a  particle  in  the  same 
line  and  in  opposite  directions  neutralize  each  other  (Art. 
10);  BO  by  this  principle  two  equal  forces  acting  in  the 
same  line  and  in  opposite  directions  at  any  points  of  a 
rigid  body  in  that  line  neutralize  each  other.  Hence  it  is 
dear  that  when  many  forces  are  acting  on  a  rigid  body, 
any  twc,  which  aie  equal  and  huvo  the  same  line  of  action 


■k$'f 


% 


ip!ii)UWUIl(«WIIUPM.IlllMIWMIlMil»ll.liHMM 


68 


SBSXTLTANT  OF  PARALLEL  FOBCKS. 


rHhM 


and  act  in  opposite  directions,  may  be  omitted,  and  also 
that  two  equsJ  forces  along  the  same  line  of  action  and  in 
opposite  directions,  may  be  introduced  without  changing 
the  circumstancQi  of  the  qrstem. 

46.  Rorahast  of  Two  PanlM 

Force*.*— (1)  Let  P  and  Q,  (Fig. 
14),  be  the  two  parallel  forces  acting 
at  the  points  A  and  B,  in  the  same 
direction,  on  a  rigid  body.  It  is  re- 
qaired  to  find  the  resultant  of  P 
and  Q. 

At  A  and  B  introduce  two  equal 
and  opposite  forces,  F.  Th^  introduction  of  these  forces 
win  not  disturb  the  action  of  P  and  Q  (Art.  44).  P  and  F 
at  A  are  equivalent  to  a  single  force,  R,  and  Q  and  /*  at  B 
are  equivalent  to  a  single  force,  8.  Then  let  R  and  S  be 
snpposed  to  act  at  0,  the  point  of  intersection  of  their  lines 
of  action.  At  this  point  let  them  be  resolved  into  their 
components,  P,  F,  and  Q,  F,  respectively.  The  two  forces, 
F,  at  0,  neutralise  each  other,  while  the  components,  P 
and  Qy  act  in  the  line  OG,  parallel  to  their  lines  of  action 
at  A  and  B.  Hence  the  magnitude  of  the  resultatti  is 
P-\-  Qt  (Art  S8).  To  find  the  point,  G,  in  which  its  line 
of  action  outs  AB,  let  the  eztremitief  of  P  and  R  (acting  at 
A)  be  joined,  and  complete  the  parallelogram.  Then  the 
triangle  PAR  is  evidently  nmikr  to  GOA ;  therefore, 

P      GO     .    .,   ,    <?       GO 
y=gj;  similarly  p=gg; 


therefore,  by  division. 


P 
Q 


OB 
GA* 


(1) 


•  Miacbte'i  BtiUlM,  |).  ». 


8CBS, 

Daifcted,  and  also 
)f  action  and  in 
itboat  changing 


£_  o_/ 


I  of  these  forces 

t.44).    Pandi^ 

1  Q  and  /*  at  B 

let  B  and  S  be 

ion  of  their  lines 

olved  into  their 

The  two  forces, 

components,  P 

lines  of  action 

the  resnltani  is 

a  which  its  line 

ind  R  (acting  at 

ram.     Then  the 

therefore, 


(1) 


RSSULTAJtT  OF  PAMAJUtML  JfOSCSS.  fift 

(2)  When  the  forces  ad  in  oppwiU  dirtctiont,^-At  A  and 
B,  (Fig  15),  apply  two  equal  and  opposite  forces  F,  as 
before,  and  let  B,  the  resultant  of  P 
and  Ft  and  S,  the  resultant  of  Q  and 
F,  be  transferred  to  0,  thiir  point  of 
intersection.  If  at  0  the  forces,  B 
and  S,  are  decomposed  into  their 
original  components,  the  two  forces 
F,  destroy  each  other,  the  force,  P, 
will  act  in  the  direetion  GO  parallel  to  the  direction  of 
Pftaah:^  and  the  force  Q  will  act  in  the  direction  OG. 
Hence  the  resultant  is  a  force  =  P  —  Q,  acting  in  the  line 
GO.  To  find  the  point  G,  we  har^  irom  the  similar 
triangles,  PAB  andwOGA, 


ris.u 


F~  QA'  "^/'-rin' 


GB' 


6B 
i9A' 


m 


Hence  the  resultant  of  two  parallel  forces^  aiding  in  the 
same  or  opposite  directions,  at  the  extremities  of  a  rigid 
right  line,  is  parallel  to  the  components,  equal  ta  tlisir 
algebraic  sum,  and  divides  the  line  or  the  line  produced^ 
into  two  segments  which  are  inversely  as  the  forces. 

In  both  cases  we  have  the  equation 

P  X  GA  =  <?  X  GR  (8) 

Hence  the  following  theorem : 

If  from  a  point  on  the  resultant  of  two  parallel  forces  a 
right  line  be  drawn  meeting  the  forces,  whether  perpendicu- 
larly or  not,  tfie  products  obtained  by  multiplying  each  force 
by  its  distance  from  the  resultant,  measured  along  the  arbi- 
traty  line,  are  equal. 

ScH.— The  point  G  poMesa^ti  this  remarkable  property ; 


'i\m 


iF** 


mmfimmmmt 


-^^f^^^Kl^n^^^S^ 


60 


MOMENT  OP  A    POBCS. 


L 


that,  however  P  and  Q  are  turned  about  their  points  of 
application,  A  and  B,  their  directions  remaining  parallel, 
G,  determined  aa  above,  remains  fixed.  This  point  is  in 
consequence  called  the  centre  of  the  parallel  forces,  P 
and  Q. 

46.  Moment  of  a  Force.— 7%e  moment  of  a  force  with 
respect  to  a  point  is  the  product  of  f  lie  force  and  the  perpen- 
dicular let  fall  on  its  line  of  action  from  th«  point.  The 
moment  of  a  force  measures  its  tendency  to  produce  rota- 
tion about  a  fixed  point  or  fixed  axis. 
Thus  let  a  force,  P,  (Fig.  16),  act  on 
a  rigid  body  in  the  plane  of  the  paper, 
and  let  an  axis  perpendicular  to  this 
plane  pass  through  the  body  at  any 
point,  0.  It  is  clear  that  the  effect  of 
the  force  will  be  to  turn  the  body  round  this  axis  (the  axis 
being  supposed  to  be  fixed),  and  the  turning  effect  will 
depend  on  the  fhagnitude  of  the  force,  P,  arid  the  perpen- 
dicular distance,  p,  of  P  from  0.  If  P  passes  through  0, 
it  is  evident  that  no  rotation  of  the  body  round  0  can  take 
place,  whatever  be  the  magnitude  of  P ;  while  if  P 
vanishes,  no  rotation  will  take  place  however  great  p  may 
be.  Hence,  the  measure  of  the  power  of  the  force  to 
produce  rotation  may  be  represented  by  the  product 

P-P, 

and  this  product  has  received  the  special  name  of  Moment. 

The  unit  of  force  being  a  pound  and  the  unit  of  length  a 
foot,  the  unit  of  moment  will  evidently  Imj  &  foot-pound. 

The  point  0  is  called  the  origin  of  moments,  and  may  or 
may  not  be  chosen  to  coincide  with  the  origin  of  co- 
oi-dinates.  The  solution  of  problems  is  often  greatly  sim- 
plified by  a  projwr  selection  of  the  origin  of  moments.  The 
perpendicular  from  the  origin  of  moments  to  the  action  line 
of  the  force  is  called  the  arm  of  the  force. 


their  points  of 
aining  parallel, 
his  point  is  in 
allel  forces,  P 

of  a  force  mth 
nd  the  perpen- 
h^  point.  The 
I  produce  rota- 


axis  (the  axis 

ing  effect  will 

nd  the  perpen- 

eo  through  O, 

nd  0  can  take 

while    if    P 

great  p  may 

the  force  to 

rodact 


e  of  Moment. 
lit  of  length  a 
)ot-pound. 
s,  and  may  or 
origin  of  co- 
greatly  sim- 
oments.  The 
he  action  line 


siaifa  OF  MOHEifra. 


61 


47.  Signs  of  Moments. — A  force  may  tend  to  turn  a 
body  about  a  point  or  about  an  axis,  in  cither  of  two  direc- 
tions; if  one  be  regarded  as  positive  the  other  must  be 
negative;  und  hence  we  distinguish  between  positive  and 
negative  moments.  For  the  sake  of  uniformity  the  moment 
of  a  force  is  said  to  be  positive  when  it  tends  to  turn  a  body 
from  left  to  right,  t.  e.,  in  the  direction  in  which  the  hands 
of  a  clock  move ;  and  negative  when  it  tends  to  turn  the 
body  from  right  to  left,  or  opposite  the  direction  in  which 
the  hands  of  a  clock  move. 

48.  Oeometrlc  Representation  of  the  Moment  of 
a  Force  with  respect  to  a  Point— Let  the  line  AJ} 
(Fig.  16),  represent  the  force,  P,  in  magnitude  and  direc- 
tion, and  p  the  perpendicular  OC  ;  then  the  moment  of  P 
with  respect  to  0  is  AB  xp  (Art.  46).  But  this  is  double 
the  area  of  the  triangle  AOB.  Hence,  the  moment  of  a  force 
with  respect  to  a  point  is  geometrically  represented  by  double 
the  area  of  the  triangle  whose  base  is  the  line  representing 
the  force  in  magnitude  and  direction,  and  whose  vertex  is 
the  given  point, 

49.  Case  of  Two  Equal  and  Opposite  Parallel 
Forcea — If  the  forces,  P  and  Q,  in  Art.  45,  (Fig.  16)  are 
equal,  the  equation 

P  X  GA  =  ^  X  GB 

gives  GA  =  GB,  which  is  true  only  when  G  is  at  infinity 
on  AB ;  also  the  resultant,  P—Q,  is  equal  to  zero.  Such  a 
system  is  called  a  Couple. 

A  Couple  consists  of  two  equal  and  opposite  parallel  forces 
acting  on  a  rigid  body  at  a  finite  distance  from  each  other. 

We  shall  investigate  the  laws  of  the  composition  and 
resolntion  of  couples,  since  to  these  the  composition  and 


62 


MMIHI 


MOMSNT   OF  A    COUP  LB. 


resolution  of  forces  of  every  kind  acting  on  a  rigid  body 
may  be  reduced. 


Fi|.l7 


V 


50.  Moment  of  a  Conpla— Let  0     ^ 

(Fig.  17)  be  anj  point  in  the  plane  of  the 

couple;   let  fall  the  perpendiculars  Oa 

and  Ob  on  the  action  lines  of  the  forces 

P.    Then  if  0  is  inside  the  lines  of  actio.i 

of  the  forces,  both  forces  tend  to  produce 

rotation  round  O  in  the  same  direction,  and  therefore  the 

sum  of  their  moments  is  equal  to 

P{Oa  +  Ob),  or  P  X  e* 

* 

If  the  point  chosen  is  O',  the  sum  of  the  moments  is 
evidently 

P  (O'a  -  0'*),  or  P  X  oi, 

which  is  the  same  as  before.    Hence  the  moment  of  the 
couple  with  respect  to  all  points  in  its  plane  is  constant. 

The  Arm  of  a  couple  is  the  perpendicular  distance 
between  the  two  forces  of  the  ooaple. 

-  The  Moment  of  a  couple  is  the  product  of  the  arm  and 
ono  of  the  forces. 

The  Axis  of  a  couple  is  a  right  line  drawn  from  any 
chosen  point  perpendicular  to  the  plane  of  the  couple,  and 
of  such  length  as  to  represent  the  magnitude  of  the  mo- 
ment, and  in  such  direction  as  to  indicate  the  direction  in 
which  the  couple  tends  to  turn. 

As  the  motion,  in  Statics  is  only  virtual^  and  not  actual, 
the  direction  of  the  axis  is  fixed,  but  not  the  position  of  it; 
it  may  be  any  line  perpendicular  to  the  plane  of  the  couple, 
and  may  be  drawn  as  follows ;  imagine  a  watch  placed  in 
the  plane  in  which  several  couples  act.  Then  let  the  axes 
of  those  coaples  whifih  tend  to  prodoce  rotation  in  the 


n  a  rigid  body 


a      o     b    d 


I 


d  therefore  the 


he  moments  is 


noment  of  the 
is  constant. 
colar  distance 

>f  the  arm  and 

»wn  from  any 
le  couple,  and 
udr  of  the  mo- 
le direction  in 

ad  not  actital, 
position  of  it ; 
of  the  couple, 
itch  placed  in 
m  let  the  axes 
tation  in  the 


COUfLKB. 


6S 


direction  of  the  motion  of  the  hands  be  drawn  upward 
through  the  face  of  the  watch,  and  the  axes  of  those  which 
tend  to  produce  the  contrary  rotation  be  drawn  tlowntmrd 
through  the  back  of  the  watch.  Thus  each  couple  is  oom- 
plotoly  represented  by  its  axis,  which  is  drawn  upward  or 
downward  according  as  the  moment  of  the  couple  is  posi- 
tive or  negative  ;  and  couples  are  to  be  resolved  and 
compounded  by  the  same  geometric  constructions  performed 
with  reference  to  their  axes  as  forces  or  velocities,  witii 
reference  to  the  lines  which  diredly  represent  them. 

Wo  shall  now  give  three  propositions  showing  that  the 
effect  of  a  couple  is  not  altered  when  cu-tain  changes  are 
made  with  respect  to  the  couple. 

51.  Tlie  Effect  of  a  Couple  on  a  Rigid  Body  is  not 
altered  if  the  arm-  be  turned  through  an;/  angle 
about  one  extremity  in  the  plane  of  the  Couple. 

Let  the  plane  of  the  paper  be  the 
plane  of  the  couple,  AB  the  arm  of 
the  original  couple,  AB'  its  new  posi- 
tion, and  P,  P,  the  forces.  At  A 
and  B'  respectively  introduce  two 
forces  each  equal  to  P,  with  their 
action  lines  perpendicular  to  the  arm 
AB',  and  opposite  in  direction  to 
each  other.  The  effect  of  the  given 
couple  is,  of  course,  unaltered  by  the  introduction  of  those 
forces.  Let  BAB'  =  W ;  then  the  resultant  of  P  acting  at 
B,  and  of  P  acting  at  B',  whose  lines  of  action  meet  at  Q, 
is  2F  sin  6,  acting  along  the  bisectw  A^ ;  and  the  result- 
ant of  P  acting  at  A  peipendicular  to  AB  and  of  P  per- 
pendicular to  AB',  is  %P  sin  9,  acting  along  the  bisector 
AQ  in  a  direction  opposite  to  the  former  resultant  Hence 
these  two  resultants  nentjalire  each  other;  and  there 
remains  the  couple  whose  arm  is  AB',  and  whose  forces  aro 
P,  P.    Hence  the  effect  flf  the  couple  is  not  altered. 


,.--m':f.x»mum\uimm 


ummmmmmi:'. 


64 


COUPLES. 


4 


\ 


^B 

Fig.  19 


52.  The  Effect  of  a  Couple  on  a  Rigid  Body  is 
not  altered  if  we  transfer  tlie  Cojtple  to  any  other 
Parallel  Plane,  the  Arm  remaining  parallel  to 
itself. 

Let  AB  be  the  arm,  antl  P,  P,  the 
forces  of  the  given  couple;  let  A'B' 
be  the  new  position  of  the  arm  par- 
allel to  AB,  At  A'  and  B'  apply  two 
eqaal  and  opposite  forces  each  equal 
to  P,  acting  pi^rpendicular  to  A'B', 
and  in  a  plane  parallel  to  the  plane  of 
the  original  couple.  This  will  not  alter  the  effect  of  the 
given  couple.  Join  AB',  A'B,  bisecting  each  other  at  0  ; 
then  P  at  A  and  P  at  B',  acting  in  parallel  lines,  and  in 
the  same  direction,  are  equivalent  to  2P  acting  at  0  ;  also 
P  at  B  and  P  at  A',  acting  in  parallel  lines  and  in  the 
same  direction,  are  equivalent  to  2/*  acting  at  0.  At  O 
therefore  these  two  resultants,  being  equal  and  opposite, 
neutralize  each  other ;  and  there  remains  the  couple  whose 
arm  is  A'B',  and  whose  forces  are  each  P,  acting  in  the 
same  directions  as  those  of  the  original  couple.  Hence  the 
effect  of  the  couple  is  not  altered. 

53.  The  Effect  of  a  Couple  on  a  Rigid  Body  is 
not  altered  if  we  replace  it  by  another  Couple  of 
which  the  Moment  is  the  same ;  the  Plane  remain- 
ing the  same  and  the  Arms  being  in  the  same 
straight  line  and  having  a 
common  extremity. 

Let  AB  be  the  arm,  and  P,  P,  the 
forces  of  the  given  couple,  and  sup- 
pose P=Q+R.  Produce  AB  to  C 
Bo  that 

AB  :  AC    ::     Q  :  P{=Q  +  B), 

and  therefore       AB  :  BO    ::    Q  :  B', 


tP=Q+R 


F{gM 


♦  P=Q+R 


(1) 


igid  Body  is 

to  any  other 

parallel  to 


\/ 


7' 


Fifl.l9 

I  effect  of  the 
!i  other  at  O  ; 
1  lines,  and  in 
]g  at  0 ;  also 
es  and  in  the 
:  at  O.  At  O 
and  opposite, 
couple  whose 
acting  in  the 
Hence  the 


^,id  Body  is 
r  Couple  of 
ne  rcniain- 
the  same 


mmm 


.30 


Q 
P=Q+R 


(1) 

(2) 


FOSCS    AND    A    COUPLB. 


66 


at  0  introduce  opposite  forces  each  equal  to  Q  and  parallel 
to  P  ;  this  will  not  alter  the  effect  of  the  couple. 

Now  E  &t  A  and  Q  at  C  will  balance  Q  +  li  at  B  from 
(3)  and  (Art.  45);  hence  there  remain  the  forces,  Q,  Q, 
acting  on  the  arm,  AC,  which  form  a  couple  whose  moment 
is  equal  to  that  of  P,  P,  with  arm,  AB,  since  by  (1)  we 
have 

P  X  AB  =  Q  X  AC. 

Hence  the  effect  of  the  couple  is  not  altered. 

Rem. — From  the  last  three  articles  it  appears  that  we 
may  change  a  couple  into  another  couple  of  equal  moment, 
and  transfer  it  to  auy  position,  either  in  its  own  plane  or 
in  a  plane  parallel  to  its  own,  without  altering  the  effect  of 
the  couple.  The  couple  must  lomain  unchanged  so  far  as 
concerns  the  direction  oj  rotation  which  its  forces  would 
tend  to  give  the  arm,  i.  e.,  the  axis  of  the  couple  may  be 
removed  parallel  to  itself,  to  any  position  within  the  body 
acted  on  by  the  couple,  while  the  direction  of  the  axis  from 
the  plane  of  the  couple  is  unaltered  (Art  50). 

54.  A  Force  and  a  Couple  acting  in  the  same 
Plane  on  a  Rigid  Body  arc  equivalent  to  a  Single 
Force. 

Let  the  force  be  /'and  the  couple  {P,  a),  that  is,  P  is 
the  magnitude  of  each  force  in  the  couple  whose  arm  is  a. 

Then  (Art  53)  the  couple  (P,  o)  =  the  couple  \F,  ^. 

Let  this  latter  couple  be  moved  till  one  of  its  forces  acts  in 
the  same  line  as  the  given  force,  F,  but  in  the  opposite 
direction.  The  given  force,  F,  will  then  be  destroyed,  and 
there  will  remain  a  force,  F,  acting  in  the  same  direction 
as  the  given  one  and  at  a  perpendicular  distance  from  it 

aP 
=  -F' 


S 


•  i 


v8 


RBSULTANT    OF   COUPLXa. 


Cob.-— J  force  and  a  couple  acting  on  a  rigid  h}dy  cannot 
produce  equilibrium.  A  couple  can  be  in  equilibrium  only 
with  an  equivalent  couple.  Equivalent  couples  are  thotie 
whose  moments  are  equal.* 

The  resultant  of  several  couples  is  one  which  will  produce 
tk'j  safne  effect  singly  as  tfie  component  couples. 

55.  To  find  the  Resultant  of  any  number  of 
Couples  acting  on  a  Body,  the  Planes  of  the 
Couples  being  parallel  to  each  other. 

Let  P,  Q,  R,  etc^  be  the  forces,  and  a,  b,  c,  etc.,  their 
arms  respoctively.  Suppose  all  the  coaples  transferred  to 
the  same  piano  (Art.  52) ;  next,  let  them  all  be  transferred  sa 
as  to  have  their  arms  in  the  same  straight  line,  and  one 
extremity  common  (Art.  51) ;  lastly,  let  them  bo  replaced 
by  other  couples  having  the  same  arm  (Art.  53).  Let  »  be 
the  common  arm,  and  P,,  Qi,  R^,  etc.,  the  new  forces, 
80  that 

P^a  ::=:  Pa,    Qia  =  Qb,     Uia  —  Re,  etc., 
then       P|  =  P-,   Qi  =  e-,   Rt=R*-,  etc., 

i.e.,  the  new  forces  are  P   ,  Q   ,  R-,  etc.,  actlug  on  the 

common  arm  a.     Hence  their  resultant  will  be  a  ooaple  of 
which  each  force  equals 

p?  4.  ^*  4-  i?*:  ^.  etc., 
a  a  a 

•nd  the  arm  ~  n,  or  tho  moment  cquaU 

Pa+  Qb  +  Re  +  etc 

If  one  of  tho  oonplcs,  as  Q,  act  in  a  direction  opposite  to 

•  The  nioDteaU  nroqiUmlent  cou|iit«  iiiajr  >wt(i  like  or  uullke  ilgiM. 


^?«?? ' 


id  body  cannot 
mlibrium  only 
pUa  are  thoKC 

h  will  produce 


number  of 
ines   of  the 

»,  c,  etc.,  their 
transferred  to 
}  transferred  sa 
',  line,  and  one 
;m  bt)  replaced 
•3).  Let  a  be 
10  new  forces, 

c,  etc., 

actliig  on  the 
le  a  couple  of 


11 


m 


1  opponito  tt 


iliko  llgM. 


aSSULTANT    OF   TWO    COUPLES. 


•Y 


the  other  couples  its  sign  will  be  negative,  and  the  foroe  at 
each  extremity  of  the  nrm  of  the  resultant  couple  jv  ill  be 

p?_^*  +  i?-  +  etc. 

Hence  the  moment  of  the  resultant  couple  is  equal  to  the 
algebraic  sum  of  the  momentfi  of  the  component  couples. 

S6.   To   Find  the  Resultant  of  two  Couplet  not 
acting  in  the  same  Plane,* 

Let  the  planes  of  the  con  pies  b^ 
inclined  to  each  other  at  an 
angle  y ;  let  the  couples  be  trejis- 
ferred  in  their  pianos  so  as  to 
have  the  same  arm  lying  along 
the  line  of  intersection  of  the  two 
planes  ;  and  let  the  forces  of  the 
couples  thus  traiisfbn-ed  Ixi  P  and  Q.  Let  AB  hk\  the  com- 
mon arm.  I^et  R  be  the  rosnltant  of  the  forces  J"'  and  Q  at 
A  acting  in  the  ^i!  rection  AJS ;  and  of  P  and  (?  at  B  acting 
in  the  direction  Jtit.  Then  einoe  P  and  Q  at  A  are  parallel 
to  P  and  Q  at  B  respectively,  tlierefore  £  at  A  is  parallel 
to  i2  at  B.  Hence  the  two  couples  are  equivalent  to  the 
single  couple  R,  R,  acting  on  the  arm  AB  ;  and  since 
PA.Q  =.  y,  V.0  LiTO 


i?  =  P«  +  g»  +  iPQ  cofi  Y  (Art  30). 


(1) 


Draw  A«,  Bi  perpendicular  to  the  planes  of  the  conples 
/•,  /',  and  Q,  Q,  respectively,  and  proportional  in  length  to 
thflir  moments. 

Draw  Ac  perpendicular  to  the  plane  of  R,  R,  and  in  the 
same  proportion  to  Aa,  Bb,  that  the  moment  of  the  couple, 
R,  R,  is  t<^)  those  of  P,  P,  and  Q,  Q,  respeotively.  Then 
Art,  Kb,  Ac,  nuiy  hi'  Inkoii  bb  the  axes  of  P,  P ;  Q,  Q;  and 

*  tMhoDlar't  autlM,  p.  «.    AtM  PraU't  UmIuuiIo*,  p.  K 


■HMMMRMP 


G8 


RESntTAXT  OF    TWO    COUPLES. 


R,  R,  respectively  (Art.  50).  Now  the  three  strai^^^ht  linos. 
Art,  Ar,  Kb,  make  the  same  angles  with  each  other  that 
AiF^  A^,  A^  make  with  each  other;  also  they  are  in  the 
euine  proportion  in  which 


or  in  which 


AB  •  P,   AB  .  i2,   AB  .  ^  are, 
P,  /?,  ^  are. 


But  R  is  the  resultant  of  P  and  Q  ;  therefore  Ac  is  tho 
diagonal  of  the  parallelogram  on  Ao,  Ai  (Art.  30). 

Hence  if  two  straight  lines,  having  a  common  extrnmitif, 
represent  the  axes  of  two  couples,  that  diagonal  of  the 
parallelogram  descril/ed  on  these  straight  lines  as  adjacent 
sides  which  passes  through  their  comn*on  extremity  repre- 
sents  the  axis  of  the  resultant  couple. 

Con. — Since  R  •  AB  is  the  axis  or  moment  of  tho  reeui'- 
ant  couple,  we  have  from  (1) 

/p.  All'  =  /".  Al?+  Q»-  A^+iP-  AB-  Q-  ABcos  y.  (2) 

If  X  and  M  represent  the  axes  or  moments  of  the  com- 
ponent couples  and  G,  that  of  the  resultant  couple,  (2) 
Dtcones 


CP  =  L*  +  M'  +  2L  •  M  COB  y. 


(3) 


Pen.  1. — \{  L,  M,  N,  are  the  axes  of  three  component 
couples  which  act  in  planes  at  right  angles  to  one  a.iother, 
and  G  tho  axis  of  the  resultant  couple,  it  may  easily  be 
shown  that 


C*=  D  +  M*^  N\ 


(4) 


If  A,  n,  V  be  the  angles  which  the  axis  of  tho  resultant 
makes  with  those  of  the  componont8,  we  have 


009  A  = 


^' 


M 

cos  /*  =  ^-, 


y 

COS  V  =  ^. 


vs. 


B  straight  linos, 
ach  other  that 
;hoy  are  in  the 


;fore  Ac  \a  tho 
rt.  30). 

mon  extremity, 
iagonal  of  the 
Ines  as  adjacent 
xtremity  repre- 

it  of  tho  resuit- 

AB-cosy.  (3) 

ta  of  the  com- 
iit  couple,  (2) 

(3) 

ree  component 
o  one  aiiothcr, 
may  easily  bo 

(*) 

tho  rosnltanf 


o' 


^ggggggm*«*- 1 1,111,11  iiiii«||p|iii.ii. 


VARroifoy's  theorem  of  moments. 


iBil"i»SBS.r 


6tf 


Sen.  2. — Hence,  conversely  any  couple  may  be  replaced 
by  three  coupIos  acting  in  planes  at  right  aiigloa  to  ono 
another ;  their  moments  being  G  cos  k,  G  cos  /u,  G  cos  v  ; 
where  G  is  the  moment  of  the  given  couple,  and  A,  p,,  v  tho 
angles  its  axis  makes  witli  the  axes  of  the  three  couples. 

Thus  the  composition  and  rcaolution  of  covplcs  follow 
laws  similar  to  those  which  apply  U)  forces,  tho  ••xis  of  the 
couple  corresponding  to  tho  direction  of  the  force,  and  the 
moment  of  the  couple  to  the  magnitude  of  tho  force. 

57.  Varignon's  Theorem  of  Moments.— TTta  mo- 

rri'CMt  of  the  resultant  of  two  conifjonen!,  forces 
with  respect  to  any  point  in  their  pic  %e  is  equal 
to  the  algebraic  sum  of  the  mnmenis  of  the  two 
components  with  respect  to  the  same  point. 

Let  A  P  and  .4  Q  represent  two  com- 
ponent forces ;  complete  the  parallelo- 
gram and  draw  the  diagonal,  Ali, 
representing  the  resultant  force.  lict 
O  be  tho  origin  of  moments  (Art.  46). 
Join  OA,  OP,  OQ,  OR,  and  draw  PC 
and  QB  parallel  to  OA,  and  let  p  =  tho  perpendicular  let 
fall  from  0  to  All. 

Now  tho  moment  of  A  P  about  0  is  the  product  of  A  P 
and  the  per|)endicular  let  fali  on  it  from  0  (Art.  40),  which 
IS  double  the  area  of  the  triangle,  AOP  (Art.  48).  But 
the  area  of  the  triangle,  AOP,  =  the  area  of  tlie  triangle, 
AOC,  since  these  triangles  have  tho  same  base,  AO,  and 
are  between  tho  same  parallels,  AO  and  CP.  Hen^o  the 
moment  of  AP  about  0  =  the  moment  of  vlC  about 
O  =  AC -p.  Also  the  moment  of  AQ  v^^.cni  0  is  dooblo 
tho  area  of  tho  triangle,  A  OQ,  =  double  the  area  of  tho 
triangle,  A  OB,  since  the  two  triangles  hovo  the  same  baso, 
AO,  end  are  between  thn  same  parallels,  AO  and  QB. 
Hence  tlie  moment  of  AQ  about  0  —  Uu'  moment  of  AU 


Fig.Zl 


70 


VASIOjrON'S  TBBOREX  OF  MOMEKTS. 


about  0  =  AB  •  p.  Therefore  the  sura  of  the  moments  of 
AP  and  AQ  about  0  =  tlie  sum  of  the  moments  ot  AC 
and  AB  about  0  =  {AC  +  AB)p,  =  {A3  +  £i?)p, 
{auci  AC  =  fift  firom  the  equal  triangles  ^PCand  QBR) 
=  AR  '  p  =  the  moment  o£  the  resulttmt. 

If  the  origin  of  momente  fall  between  AP  and  AQ,  the 
forces  will  tend  to  produce  rotation  in  opposite  directions, 
and  hence  their  moments  will  have  contrary  signs  (Art 
4t).  In  this  caae  the  moment  of  the  resultant  =  the  dif- 
ference of  the  moments  of  the  components,  as  the  student 
will  find  no  difficulty  in  showing.  Hence  in  either  case 
the  moment  of  the  resultant  is  equal  to  the  algebraic  sum 
of  the  moments  of  the  components. 

Con.  1.— If  there  are  any  number  of  component  forces, 
wo  may  compound  them  in  order,  taking  any  two  of  them 
first,  then  finding  the  resultant  of  these  two  and  a  third, 
and  so  on ;  and  it  follows  that  the  sum  of  their  momenta 
(with  their  proper  signs),  is  equal  to  the  moment  of  the 
redultojit. 

OoR.  2.— If  the  origiv  of  moments  be  on  the  line  of 
action  of  the  reroltant,  p  =  0,  and  therefore  the  moment 
of  the  resultant  =  0  ;  hence  the  sum  of  the  moments  of 
the  componeuta  is  ecjual  to  isero.  In  this  case  the  moments 
of  the  forces  in  one  direction  balance  those  in  the  opposite 
dirpction ;  t.  e,,  the  forces  that  tend  to  produce  rotation  in 
one  direction  Mf^  counteracted  by  the  forces  that  tend  to 
produce  rotation  in  the  opposite  direction,  and  there  is  no 
tenden'jy  to  rotation. 

Oo».  3.— If  all  the  forces  are  in  eqailibriam  the  resultant 
J?  =  0,  and  therefore  the  moment  oi  R  —  Q;  henoe  the 
sum  of  the  momontfl  of  the  conponents  is  equal  to  aewj, 
and  there  is  no  teudeacty  to  motion  either  of  tnuiulatiou  or 
rottktion. 


VlfTS. 

the  moments  of 
loments  ot  AC 
{A3  +  BB)p, 
IPC  and  QBJi) 

iP  and  AQ,  the 
>aite  directions, 
rary  signs  (Ai-t. 
[«nt  =  the  dif- 
I  as  the  student 
i  in  either  case 
3  alj/ebratc  sum 


nponent  forces, 
,ny  two  of  them 
,\vo  and  a  third, 
their  moments 
moment  of  the 


on  the  line  of 
re  the  moment 
he  moments  of 
se  the  moments 
in  the  opposite 
uoe  rotation  in 
[>s  that  tend  to 
md  there  is  no 


im  the  resultant 
=  0;  hence  the 
equal  to  serri, 
trauulatiou  or 


p 

R 

Q 

1 

( 

1 

B 

VARIGNOIf'a  TBEORMM  FOR  PAKALLSL  FORCES.     71 

CoK.  4. — Therefore  when  the  moment  of  the  resultant 
=  0,  we  conclude  either  that  the  resultant  —  0  (Cor.  3), 
or  that  it  passes  through  the  point  taken  aa  the  origin  of 
moments  (Cor.  2). 

58.  Varignon'B  Tlieoram  of  MomamtB  for  Parallel 
Forcea. — The  aum  of  the  rnamients  of  two  parallel 
forces  abotit  any  point  is  equal  to  the  moment  of 
their  resultant  about  the  point. 

Let  P  and  Q  be  two  paraliel  forces 
acting  at  A  and  B,  and  R  their  result- 
ant acting  at  6,  and  let  0  be  the  point 
about  which  moments  are  to  be  taken. 
Then  (Art.  45)  we  have 

P  X  AG  =  Q  X  BG, 
.•.    P(OG  -  OA)  =  C  (OB  -  OG), 
.-.       (P  +  C)  OG  =  P  X  OA  +  C  X  OB, 

/Z  xOG  =  PxOA  +  G  xOB; 

that  is,  the  earn  of  the  moments  =  the  moment  of  the 
resultant 

OoB.— It  follows  that  the  algebraic  sum  of  the  momenta 
of  any  number  of  parallel  forces  in  one  plane,  with  respect 
to  a  point  in  their  plane,  is  equal  to  the  moment  of  their 
resultant  with  respect  to  the  point 

69.  Centre  of  Parallel  Forcea.  -To  iind  ttte  mag- 
nitude, direction,  and  point  of  application  of  the 
resultant  of  any  number  of  parnllel  forces  acting 
on  a  rigid  body  in  one  plane. 


kHH 


iT'lMMBtmill 


72 


CBNTRV  OF  PARALLEL  FORCES. 


Let  P.,  P,,  P.,  etc.,  denote  the 


ZM 


fonjM,  if,,  Jf,,  M3,  etc.,  their  points    ^      -.       >■''• 
of  ufplicution.     .Take   any  point  in 
the  plane  of  the  forces  as  origin  and 
draw  the  rectangular  axes  OX,  OV. 

L«t  («i.  yi).  (««»  1/%)f  etc.,  be  the 
points  of  application,  if,,  i/,,  etc. 
Join  M^M^;  and  take  the  point  if  on  M^M,,  so  that 


^. 


/• 


6  «& 
Fig.24 


if,  if 


(1) 


then  the  rcsnitant  of  P,  and  P,  is  P,  +  Pg,  and  it  acts 
through  M  parallel  to  P,  (Art  46). 

Draw  M^a,  Mb,  M^c  parallel,  and  Mie  perpendicular  to 
the  axis  of  y.    Then  wo  have 


p 

Mb-yy=  -jr—j-  iy,  -  Vi) '. 
,  ,     Mb-       p^^p^      , 


(2) 


which  gives  the  ordinate  of  the  point  of  application  of  the 
rcEultant  of  P,  and  Pg. 

Now  since  the  resultsat  of  P,  and  P,,  which  is 
P,  4-  P,,  acts  at  M,  the  resultant  of  P,  +  P,  at  M,  and 
P,  at  M^,  is  P,  +  P,  -f-  P,  at  g,  and  substituting  in  (8) 
P,  +  /'„  P„  ifi,  and  y,  for  P,,  P„  y,,  and  y,  rcgpec- 
tiywiy,  we  have 


mm^ 


mm 


mmt 


m. 


7m  ^ 

i^ 

g-^U, 

a  c 

»  b   0  b 

Fig.24 
r,,  BO  that 

0) 

Pg,  and  it  acts 
erpendicalar  to 


« -yi); 


y». 


(2) 


ieation  of  the 

Pg,   which    is 
I\  at  if,  and 

jtnting  in  (3) 
md  yg  ro§pec- 


ItJ-  P^Vi . 


'•  +  ^3 


;(3) 


CBHTRS  or  PARALLSL  P0BCE8, 


T3 


and  this  process  may  be  extended  to  any  number  of  parallel 
forces.  Let  R  denote  the  resultant  forCe  and  y  the  ordi- 
nate of  the  point  of  applicjation  ;  then  we  have 


i?  =  Pi  +  P,  +  ^3  +  etc. 


SP. 


7,  -  P^Vx  +  ^«y«  +  ^sy._+_.etc.  _  I.Py 
y  -        p,  +  p,  +  P3  +  etc.  ^P 


IP 


Similiirly,  if  x  bo  the  abscissa  of  the  point  of  application  of 
the  resultant,  we  have 


X  = 


XP' 


The  values  of  zl  y  are  indeponth^nl  of  the  angles  which 
the  directions  of  the  forces  muko  wllh  the  axcM.  Hence 
t'  these  directions  be  turned  about  the  points  of  application 
(f  the  forces,  their  parallelism  being  preserved,  tlie  point  of 
ii|)pliciition  of  the  resultant  will  not  move.  For  this  reason 
tlie  point  (i,  y)  is  called  the  centre  of  parallel  forces.  We 
shall  hereafter  have  many  applications  in  which  its  position 
is  of  great  importance. 

ScH.  1. — The  moment  of  a  forcf.  toifh  respect  to  a  plane 
to  which  it  is  parallel,  is  the  product  of  the  force  into  ♦ho 
perpendicular  distance  of  its  point  of  application  from  the 
plane.  Thui,  Pi^i  is  the  moment  of  the  force  P,,  iu 
reference  to  the  plane  through  OX  perpendicular  to  OV. 
This  must  be  carefully  distinguished  from  the  moment  of  a 
force  with  respect  to  a  point.  Hence  the  Aquations  for 
determining  the  posit'on  of  the  centre  of  pfprailel  forcet 
show  that  the  sum  of  tlie  mmufnts  of  the  parallel  fofoen  icitti 
respect  to  any  plane  parallel  to  tfifun,  i«  eq</0X  to  the  moment 
of  their  resultant. 

SoH.  2.  —  The  moment  of  a  force  with  respect  to  any  line 
is  the  product  of  the  component  of  the  force  |.Hirj)endicalar 


■^ 


74 


comurroxa  of  xquiLiasivx. 


♦Pt 


FlgJI 


♦r 


to  the  line  into  the  shortest  distance  between  the  line  and 
the  line  of  action  of  the  force. 

60.  Conditions  of  Equilibrium  of  a  Rigid  Body 
acted  on  by  Pandlal  Foroes  in  one  Plane.— Ut 
P,,  P„  P„  etc.,  denote  the  forces.  Take 
any  point  in  the  plane  of  the  forces  as 
origin,  and  draw  rectangular  axes,  OJT, 
OV,  the  latter  parallel  to  the  foroes.  Let 
A  bo  the  point  where  OJT  meets  the  direc- 
tion of  P,,  and  let  OA  =  x^. 

Apply  at  0  two  opposing  forces,  each 
equal  and  parallel  to  P,;  this  will  not  disturb  the  equili- 
brium. Then  P,  at  A  is  replaced  by  P,  at  0  along  OV, 
and  a  couple  whose  moment  is  P^  ■  OA,  i.  e.,  P,a;,.  The 
remaining  forces,  P„  />„  etc.,  mr.y  be  treated  in  like  man- 
ner. We  thus  obtain  a  stt  o1  forces,  P^,  P„  P„  etc., 
acting  at  0  along  OY,  and  a  sot  of  couplr..,  P^r^,  P,x,, 
/'jij,  etc.,  in  the  plane  of  the  forces  tending  to  turn  the 
^fcbdy  from  die  axis  of  x  to  the  axis  of  ;/.  These  forces  are 
etjuivalcnt  to  a  single  resultant  force  /'^  f\  +  /*,  +  etc., 
and  the  couples  iirc  equivalent  to  a  s^gji  i^ciilt&n!  rioaj^ 
Pjjr,  +  P,x,  +  P,«,  +  etc.  {Art  6^. 

Hence  denoting  the  resultant  force  by  72,  and  tlie  mumout 
of  the  resultAiil  couple  by  0,  we  have 


il  =  P,  -I-  P,  4-  /»,  -H  eta  =r  SP; 

0  =  P,*,  +  P,a!,  +  P,x,  4-  eta  .-=  S.PTi 

that  is,  a  system  of  parallel  forces  can  be  reduced  to  a 
single  fiitj*  Hiiij  a  couple,  which  (Art.  fl4.  Cor.)  (annot 
pioduoe  equilibrium.  Heuoo,  for  equilibrium,  the  force 
and  the  couple  must  vanish  ;  or 

XP  =  0,    and    LPx  =  a 


TM. 

^n  the  line  and 


B  Rigid  Body 
18  Plane.— Let 


♦P. 


turb  the  equili- 
it  0  along  OF, 
e.,  P,a;,.  The 
ied  in  hko  man- 

'        >»   ■*  S'  6tc., 

rig  10  turn  (he 

Phese  forces  are 

/%  +  /*,  + etc., 

■oopte, 

uU  the  momt)ut 


IP; 

reduced  to  n 

Cor.)  runnnf 

uin,  the  force 


V 

1 

^1      >R 

H 

'^    ,yi 

-A 

' 

0 

M' 

F 

i8.a« 

coifDmoira  ojc  squiLiBBiuM.  75 

Hence  the  ooinditions  of  eqailibriam  of  a  system  of  pur- 
iillel  forces  acting  on  a  rigid  body  in  one  pkue  are  : 

The  sum  of  the  forces  must  =  0. 

The  suin  of  the  momenta  of  the  forces  about  every  point  in 
their  plane  mttst  =  0. 

61.  Conditioiift  of  EqaililMiiiin  of  a  Rigid  Body 
acted  on  by  Forces  in  any  direction  in  one  Plane.— 

Let  Pj,  Pg,  Pj,  etc,  be  the  forces  acting  at  the  points 

(^i»  yi)>  (a^8.  y«)«  i^i'  ys)>  etc.,  in  the 
plane  xy.  Resolve  the  force  P,  into  t>Yo 
components,  X,,  F,,  parallel  to  OJT 
and  OY  respectively.  Let  the  direc- 
tion of  Kj  meet  OX  at  M,  and  the 
direction  of  X,  meet  OFat  iV.  Apply 
at  0  two  opix)sing  forces  each  equal  and  parallel  to  Xj, 
and  also  two  opposing  forces  each  equal  and  parallel  to  F, . 
Hence  Fj  at  .4,,  or  M,  is  equivalent  to  Y^  at  O,  and  a 
louplo  whoso  moment  is  F,  •  OM;  and  X^  at  J,,  or  iV,  is 
equivalent  to  Xj  at  0,  and  a  couple  whose  moment  is 

V,    ov. 

Ifenc.  Fj  is  replaced  by  F,  at  0,  and  the  couple  F,.t,  ; 
and  A",  is  voplanul  by  Xj  at  (),  and  the  couple  X|//i  (Art. 
47).  Therefore  the  force  P,  may  bo  replaced  by  the  com- 
ponents Xj,  F,  acting  at  0,  and  the  couple  whose 
moment  is 

and  which  equals  the  moment  of  P,  about  0  (Art.  67). 

By  a  similar  resolution  of  all  the  forces  we  shall  have 
them  replaced  by  the  forces  (X,,  F,),  (Xj,  Fj),  etc., 
acting  at  0  along  the  axes,  and  the  couples 

TtX,  —  X,y„     Fa^s  —  Xj^,,  etc 

Adding  together  the  couples  or  moments  of  P,,  P,,  etc., 


■i; 


76 


SQVILIOBr'M    VNDEH     THREE    FORCES. 


and  denoting  by  G  the  moment  of  tho  resultant  conple,  wc 
get  the  total  moment 

If  the  sum  of  the  cumponents  of  the  forces  along  OX  is 
denoted  by  SX,  and  the  sum  of  the  components  along  OV 
by  £F,  the  resultant  of  the  forces  acting  at  0  is  given  by 
the  equation 

i?  =  (SX)»  -h  (£F)«. 

If  a  be  the  angle  which  R  makes  with  the  axis  of  X,  wc 
have 

R  cos  a  =  IX,    Ji  Bin  a  =  I.V; 


tan  a  = 


iX' 


Therefore,  any  system  of  forces  acting  in  any  direction 
in  one  plane  on  a  rigid  botly  may  be  reduced  to  a  single 
force,  li,  and  a  single  couple  whose  moment  is  O,  which 
(Art,  64,  Cor.)  cannot  produce  equilibrium.  Hence  for 
equilibrium  we  must  have  R  =  0,  and  G'  =  0,  which 
requires  that 

tx=o,  ir=o, 

^irx  —  Xy)  =  0. 

Hence  the  conditions  of  equilibrium  for  a  system  of 
forces  acting  in  any  direction  in  one  plane  on  a  rigid  body 
arc : 

Tlfte  sum  of  the  components  of  the  forces  parallel  to  each  of 
two  rectangular  axes  ..ivM  =  0. 

The  sum  of  the  mofir  its  of  the  forces  round  ever tf  point  in 
their  plane  must  =  0. 


rt*i 


FORCES. 

ultant  coaple,  wc 


EXAMPLES. 


77 


roes  along  OX  is 
inents  along  OV 
at  0  is  givon  by 


;he  axis  of  X,  we 


in  any  direction 

iced  to  a  single 

ent  is  O,  wliich 

Hence  for 

O  —  0,   wliich 


m 


3r  a  system  of 
m  a  rigid  body 

rallel  to  each  of 
i  every  point  in 


Cor. — Convoraely,  if  the  forces  are  in  equilibrinni  tho 
sum  of  the  com{K>nents  of  the  forces  parallel  to  any  diroc- 
tioii  will  =  l>,  and  also  the  sum  of  the  moments  of  the 
forces  about  any  point  will  =  0. 

62.  Condition  of  Equilibrium  of  a  Body  under  the 
Action  of  Three  Forces  in  one   Plane  -//  thn 

forces    jtiaintain    a     hi'dy    in    equilibrium,,    their 
directions  must  meet  in-  a  point,  or  be  parallel. 

Suppose  the  directions  of  two  of  the  forces,  P  and  Q,  to 
meet  at  a  point,  and  take  moments  round  this  point ;  then 
tlio  moment  of  each  of  these  two  forces  =  0;  thcrefon>  ilio 
moment  of  the  third  force  li  =  0  (Art.  61,  Cor.),  which 
requires  either  that  Ji  —  0,  or  that  it  pass  through  the 
point  of  intersection  of  P  and  Q.  If  R  is  not  =  0,  it  must 
jtiiss  through  this  jmint.  Hence  if  any  two  of  the  forces 
meet,  tiie  third  must  pass  through  their  point  of  intersec- 
tion, an  keep  it  at  rest,  and  each  force  must  be  equal  and 
opposite  to  the  resultant  of  the  other  two.  If  the  angles 
])etween  them  in  pairs  I  c  p,  q,  r,  tho  forces  must  satisfy  the 
conditions 

r  :  Q  '.  R  =  «inp  :  sin  &  :  sin  r  (Art.  32). 

If  tro  of  the  forces  are  parallel,  the  third  must  bo 
parallel  o  them,  aud  equal  md  directly  opposed  to  their 
resultant. 

EXAMPLES.       . 

1.  Suppose  six  parallel  forces  proportional  to  the  numbers 
1,  2,  3,  4,  5,  6  to  act  at  points  (—2,  —1),  (—1,  0),  (0,  1), 
(1,  2),  (2,  3),  (3,  4) ;  find  the  resultant,  R,  and  the  centre 
of  parallel  forces. 

By  Art  69  we  have 

J2  =  SP  =  l  +  2  +  ...6  =  21; 


i 


.v^«ari«W«MaH*ea«^^j^j,^j^. 


^^Wis^^^l^^^j^i 


i=i5te*S*iS6ii6'&i^.^^;.o  , 


■^ 


J 


^%. 


,^'t*'„  w. 


\^>1' 

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CIHM/ICMH 

Microfiche 

Series. 


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Collection  de 
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^ 


.  .^aea'iifcii<a5awBfe^'"-is? 


■-  f .-.sii^irTp  tf? 


y^^^K^^^^^it'^^^^^v^^^J^^Svv^svJ.J^S' 


^i^^aSmS^ 


wsm 


•ft  XXAMPLMS, 

SP*  =-2-2  +  4  + 10 +18  =  28; 

XPy  =  -  1  +  3  +  8  +  15  +  24  =  49. 

.      i-^Z^i  -  2S.    -_2Py  _  49 
''        ~    £-"    ~21'    ^~"£F~2i' 

2.  At  the  three  vertices  of  a  triangle  pnrallel  forces  are 
applied  which  are  proportional  respectively  to  the  opposite 
Bides  of  the  triangle;  f  nd  the  centre  of  these  forces. 

Let(»i»  !ft)>  (»i,  y»),  («„  Jf,)  be  the  vertices,  and  let  a, 
i,  c  be  the  sides  opposite  to  them;  then 

;_aa;,+te,+cr,      -_  ay, +»»-,+ gy, 
a+4+c       '    »  a+i+c 

3.  If  two  parallel  foroeu,  P  and  ^,  act  in  the  same  direc- 
tion at  A  and  5,  (Pig.  ji),  and  make  an  angle,  B,  with 
^5,  find  the  moment  o/  each  about  the  point  of  epplicur 
tion  of  their  resnltant 

The  moment  of  F  with  respect  to  (7  is 

P-^(?8iu(!»(Art.46). 
But  from  (1)  of  Art  4ft,  we  have 


P+  Q 


.'.    AG  = 


4B 
AG* 


P  +  Q 


aB, 


which  in  P  -  AG  an  0  gives 


for  the  moment  of  P  which  also  e<]uals  the  moment  of  (j. 


lOfflHP^:- 


(  =  49. 


y 


49 
21* 


■nrallel  forces  are 
J  to  the  opposite 
Bse  forces, 
iiidoe^  and  let  a, 


k  the  Mune  direc- 
u  ao^,  0,  with 
point  ol  epplicur 


Qoment  of  (j. 


MXAMPtML 


n 


I 


Fit-V 


4.  Two  paiallel  foroet,  aotiBg  in  ike  nm»  direction; 
liave  their  magnitndes  5  and  13,  and  th«r  points  of  appUcar 
tion,  J  and  B,  6  feet  tipmti.  Find  the  nu^iind«  of  their 
resultant,  and  the  point  of  application,  0. 

Ant,  B  =  18,  J.0  rts  4|,  BG  =  If 

6.  On  a  Ftraif^ht  rod,  AF,  there  are  snspended  a  weights 
of  6, 15,  7,  6,  -uid  9  poands  respectively  at  the  points  A,  B, 
D,  E,  F;  AB  =  3  feet,  BD  =  Q  feet,  DE  =  &  feet, 
EF  =  4  feet  l^'ind  the  magnitnde  of  the  resultant,  and 
the  distance  of  its  point  of  application,  O,  from  A. 

Ana.  Ji  =  i2  pounds.  AG  =  6f  feet 

6.  A  heavy  nniform  beam,  AB,  rests 
in  a  vertical  phuae,  with  one  end.  A,  on  a 
emooth  horizontal  plane  and  the  other 
end,  B,  against  a  smooth  vertical  wall ; 
the  end.  A,  is  prevented  from  sliding  by 
a  horizontal  string  of  given  length  ii£- 
tened.  to  the  end  of  the  beam  and  to  the  vail ;  determine 
the  tension  of  the  string  and  the  preesurss  against  the 
horizontal  plane  and  the  wall 

Let  ito  =:  the  length  of  the  beam,  and  let  IF  be  its  weight, 
which  as  the  beam  i>  uniform,  wn  may  suppose  to  act  at  its 
middle  point,  G.  Let  B  be  the  vertical  pressure  of  the 
hori«>ntal  plauvj  against  the  beam ;  '\nd  R  the  horizoot'U 
pressure  of  the  vertical  wall,  and  T  the  tension  of  the  hor- 
izontal string,  AC ;  let  BAG  =  o,  a  known  angle,  since 
the  lengths  of  the  boaa  and  th«  string  are  given.  Then 
(Art  61),  we  have 

tot  horiaontal  forces,  T=s  R; 

for  vertioal  forces,      17  =  JK ; 

for  moments  abont  A  (Art  47),  2/2*^  dn  «  =  TTa  cos  •> ; 


80 


XXAMPLSa. 


i 


7.  A  heavy  beam,  AB  =  2a,  rests 
on  two  givea  amooth  planes  which  are 
inclined  at  angles,  a  and  /3,  to  the 
horizon ;  required  tho  angle  0  which 
the  beam  makes  with  the  horizontal 
plane,  and  the  presfiores  on  the 
planes. 

Let  a  and  b  be  the  segmeats,  AG  and  BO,  of  the  beam, 
made  by  its  centre  of  gravity,  G;  let  E  and  R'  be  tho 
pressures  on  the  planes,  AC  and  BC,  the  lines  of  action  of 
which  are  perpendicular  to  the  planes  since  they  are  smooth, 
ard  let  W  be  the  weight  of  the  beam.    Then  we  have 

for  horizontal  forces,  Rsin  a  =  R'  mnP;      \   (1) 

for  yertical  f oroM,  R  coa  a  +  R"  coa  p  =  W;      (8) 

for  moments  about  G,  ^ cos  (a—0)=:R'b cos  {li+0)\ [  (3) 

pi.riding  (3)  by  (1),  we  have 

aoot  a  -{-  atanO  =  bcotp  —  hta,n6; 

a  cut  a  —  6  cot  i? 


therefore, 


tan9  = 


a  +  b 


and  from  (1)  and  (2)  wo  have 
Wain  (3 


R  = 


sin  {a +  13)' 


and  R' 


IT  sing 
sin  (a  4-0)' 


Otherwise  thus :  since  the  beam  is  in  equilibrium  under 
the  ftotion  of  only  three  forces,  they  must  meet  in  a  point  0, 
(Art.  62),  and  therefore  we  obtain  immediately  from  the 
geometry  of  the  figure^ 


P 

w 


sing 

siu  (c+/3)' 


«  = 


Wainfi 
mi  (a  +  0)' 


IG,  of  the  beam, 
t  and  R'  be  the 
lines  of  action  of 
they  are  smooth, 
;n  we  have 

ramU;     \   (1) 

ip=  W;      {%) 

Aoo8(/3-fe),  (3) 

tan  $; 


Bin  a 

lilibrium  under 
Bet  in  a  point  O, 
lately  fh)m  the 

sin  /3 


«  +  /3)* 


■^ipBHMK^'^ 


SXAMPLSa. 


81 


ana     -n-,  = 


R 
W 


sin  a 


R'  = 


IT  Bin  a 


sin  (a  +  (iy  •    "  —  gin  („  ^  py 

Also  since  the  angles,  GOA  and  OOB,  ai-e  equal  to  a  and  (i, 

respectively,  and  AGO  =  5  —  (?,  we  have 

« 

(a  +  ft)  cot  AGO  =  o  cot  GAO  —  b  cot  GOB; 

a  cot  a  —  ft  cot  /3 


therefore, 


tan0  = 


a  +  b 


Hence,  if  x  =  K^r-gi  the  beam  will  rest  in  a  horizontal 
position. 

8.  A  heavy  uniform  beam,  AB,  rests  with 
one  end,  A,  against  a  smooth  vertical  wall, 
and  the  other  end,  B,  is  fastened  by  a  string, 
BO,  of  given  length  to  a  point,  C,  in  the 
wall ;  the  beam  and  the  string  are  in  a  vertical 
l)lane ;  it  is  required  to  determine  the  pressure 
against  the  wall,  the  tension  of  the  string,  and 
the  position  of  the  beam  and  the  string. 

Let        AG  =  GB  =  a,    AC  =  x,    BC  =  ft. 


F)|.M 


weight  of  beam  =  W,  tension  of  string  =  T,  prousare  of 
waU  =  Ji, 

BA£  =  0,    BOA  =  t. 
Then  wo  have 

for  horiiontttl  forces,  R  =:  Tain  ^;  (1) 

for  vertical  forooB,       If  =  7*  cos  ^;  (2) 

for  momenta  about  A,  Ifa  sin  9  =  T-  AD  =  Tz  sin  ^;  (%) 

. ' .    a  sin  9  =  a;  tan  9 ;  (4) 


89 


SXAMPLSa, 


and  by  the  goometr/  of  the  figure 

sin  B 


2a 


sin  ^ ' 


z  _  sin  (O—ji) 
Ha"      sin  ^ 


(8) 

(6) 


Solving  (4),  (6),  and  (6),  we  get 


.  .     1  ri6o»  -  ft»"]* 


nn 


from    which    ^   and    T  become  known.    (Price's  Anal. 
Moch'a,  Vol.  I,  p.  69). 

To  determine  M  tho  anknown  qoantitles  many  problems  in  Stat  cs 
require  eqoatlona  to  be  formed  bj  geometric  relations  as  well  as  ttatic 
relations.  Thus  (1).  (S),  (8)  are  stntic  equations,  and  (5)  is  a  geometric 
equation. 

0.  A  uniform  heavy  beam,  AB  =  2a, 
rests  with  one  end,  A,  against  the  inter- 
nal surface  of  a  smooth  hemispherical 
bowl,  radius  =  r,  while  it  is  supported 
at  some  point  in  its  length  by  the  edge 
of  the  bowl ;  find  the  position  of  equili- 
brium. 

The  beam  is  kept  in  equilibrium  by  three  forces,  vias.,  the 
reaction,  R,  at  A  perpendicular  to  the  surface  of  contact, 
(Ari  42)  and  therefore  perpendicular  to  the  oowl,  the 
reaction,  R',  at  C  which,  for  the  8an)e  reason,  is  perpen- 
dicular to  the  beam,  and  the  weight  IK  acting  at  0. 


rtg.so 


ti 


w 


01 


(1 

0 
k 

.S( 

li 


t1 
E 


(5) 

(6) 


(Price's  Anal. 

problems  in  Stat  cs 
ins  M  well  as  tttUic 
id  (6)  is  a  ge(HU«tric 


na.30 


forccB,  via!.,  the 
irface  of  contact, 
the  bowl,  the 
Mon,  is  perpon- 
ng  at  O. 


MXAMPLSa. 


83 


Let  0  =  the  inclination  of  tho  beam  to  the  horizon 
=  <AGD.  The  solution  will  be  most  readily  effected  by 
resolving  the  forces  along  the  beam  and  tbking  moments 
about  0,  by  which  we  shall  obtain  equations  free  from  the 
unknown  reaction,  R.    Then  we  have 


for  f 01*068  along  AB,  £  cos  d  =  TF  sin  0, 

for  moments  about  0, 

R'2roofi9ma.e  =  IF  (2r  cos  0  —  a)  cos  0. 

From  (1)  we  have 

/2=irtanfl, 

which  in  (2)  gives,  after  reducing, 

8r  sin*  0  —  2/*  cos*  tf  4  o  OM  0  =  0, 

4r  cos*  0  —  a  cos  0  —  2r  =  0, 


(1) 

(2) 


(ir, 


(3) 


COS*  = 


a  ±  \/32r»  +  a* 
8r 


Otherwise  thns:  since  the  beam  is  In  equilibrium  under 
I  lie  action  of  only  three  forces,  they  must  meet  in  a  point 
()  (Art.  62).  Draw  the  three  forces  AO,  CO,  GO,  which 
keep  the  beam  in  equilibrium.  Let  the  line,  00,  meet  the 
semicircle,  DAO,  in  the  point,  Q.  Then  AQ  is  a  horizontal 
line.    Also 


<QAG  =  <DCA  =  9, 

therefore 

<OAQ  =  29. 

llenoe 

AQ  =  AO  cos  20, 

:ind  also 

AQ  =  AO  cos  0; 

M 


MXAMPLXS. 


therefore 


or 


Sr  COB  20  =  a  ooa  d, 
4r  COB*  6)  —  a  COB  0  —  2r  =:  0, 


which  is  the  same  as  (3)  obtained  by  the  other  method. 
The  student  may  prove   that   the   reaction,    B',  at  G 
a 


=  »F 


2r 


n^ 


10.  Find  the  position  of  equilibrium  of 
a  uniform  heavy  beam,  on<}  end  of  which 
rests  against  a  smooth  vertical  plane,  and 
the  other  against  the  internal  surface  of  a 
smooth  spherical  bowL 

The  beam  is  in  equilibrium  under  the 
action  of  three  forces,   the  weight,  fF, 
acting  at  G,  the  reaction,  R,  at  A,  perpen- 
dicular to  the  surface  and  henoe  passing  through  the  centre, 
C,  and  the  reaction,  R',  of  the  vertical  piano  perpendicular 
to  itself  and  hence  horizontal 

Let  the  length  of  the  beam,  AB,  =  2fl,  r  =  the  radius 
of  the  Bphere,  d  =  CD,  the  distance  of  the  centre  of  the 
sphere  from  the  vertical  wall,  W  =  the  weight  of  the  beam ; 
and  let  6  =  the  required  inclination  of  the  beam  to  the 
horiion,  and  ^  =  the  inclination  of  the  radius  AO  to  the 
horixon.    Then  we  have 

for  rertioal  forces,  /2  sin  ^  =.  fT ;  (1) 

for  moments  about  B,  R-  2a  sin  (^—9)  =  W-aootO;  (8) 

Dividing  (2)  by  (1)  we  have 

a  sin  (»  —  0) 


or 


BlU  ^ 

tan  ^  =  2  tan  #. 


008  9, 


(3) 


mrmi 


0, 

her  method, 
ction,    B',  at  G 


n«.ii 


rough  the  centre, 
10  perpeudicnlar 

,  r  =  the  radius 
he  centre  of  the 
ht  of  the  beam ; 
le  beam  to  the 
adius  AO  to  the 


W;  (1) 

W-ao(me;  (3) 


(3) 


CSlfTBS    OF  PARALLEL    FORCKS. 


85 


Then  we  ha'^e,  from  the  geometry  of  the  fignre,  the 
liorizontal  distance  from  A  to  the  wall  —  the  horiiontal 
projection  of  AO  4-  CD,  that  is. 


fta  coa  9  z=  r  COS  <f>  ■{•  d. 


(4) 


From  (3)  and  (4)  a  value  of  6  can  be  obtained,  and  hence 
tlie  position  of  equilibrium. 

Other^'ise  thus :  since  the  beam  is  in  equilibrium  under 
tiie  action  of  only  three  forces  they  must  meet  in  a  point,  0. 
Geometry  then  gives  us 

2  cot  0GB  =  cot  AOG  —  cot  GOB  =  cot  AOG, 


or 


2  tan  9  =  tan  ^ 


which  is  the  same  as  (3). 

63.  Centre  of  Parallel  Forces  in  Different  Planes. 
— To  find  the  magnitude,  direction,  and  point  of 
application  of  the  resultant  of  any  number  of 
parallel  forces  dieting  on  a  rigid  body. 

The  theorem  of  Art.  59  is  evidently  true  also  in  the  cose 
in  which  neither  the  parallel  forces  nor  their  fixed  points  of 
application  lie  in  the  same  plane,  hence,  calling  $  the  third 
co-ordinate  of  the  point  of  application  of  the  resultant,  we 
have  for  the  distance  of  the  centre  of  parallel  forces  from 
tlie  planes  yz,  zx,  and  xy, 


m  = 


:lPx      -  _  I;Py      .  _  1P« 


iF 


y  = 


ii* 


£P' 


Hence  (Art.  59,  Sch.)  the  equations  for  determining  the 
position  of  the  centre  of  parallel  forces  show  that  the  sum 
of  the  f I' omenta  of  the  parallel  forces  with  respect  to  any 
plane  paraiUi  to  litem  is  equal  to  the  mommt  of  their 
reanUani. 


li'i 


»^^ 


iiiiiJMiaiiiliiiiuii^^ 


•^  V 


Fis*31 


86  squtLTBRum  of  parallsl  forcbb  m  spacxs. 

64.  Conditioiu  of  Equilibrinm  of  a  Sjrstem  of 
Parallel  Forces  Acting  upon  a  Rigid  Body  in 
Space. — Let  P,,  P„  P,,  etc.,  denote  the  forceF,  and  let 
them  be  referred  to  three  rectangular  axes, 
OX,  or,  OZ;  the  last  parallel  to  the 
forces  ;  let  (a;,,  y„  «,),  (z„  y,,  «,),  etc., 
be  the  points  of  application  of  the  forces, 
/•,,  Pj,  etc.  Ijet  the  direction  of  P^ 
meet  the  plane,  ay,  at  Af,. 

Draw  if,i\r,  perpendicular  to  the  axis 
of  X  meeting  it  at  JV,.  Apply  at  O,  and  also  at  JV,,  two 
opposing  forces  each  equal  and  parallel  to  P^.  Then  the 
force  Pj  at  Mi  is  replaced  by 

(1)  P,  at  0  along  OZ; 

(2)  a  couple  formed  of  P,  at  Jf,  and  P,  at  JV, ; 

(3)  a  couple  formed  of  P,  at  JV,  and  P,  at  0. 

The  moment  of  the  first  couple  is  Pijft,  and  this  couple 
may  be  transferred  to  the  plane  ye,  which  is  parallel  to  its 
original  plane,  without  altering  its  effect  (Art.  52).  The 
moment  of  the  second  couple  is  P,a;,,  and  the  couple  is  in 
the  plane  xz. 

Replacing  each  force  in  this  manner,  the  whole  ejstem 
will  be  equivalent  to  a  force 

■Pi  +P,  +  Ps +etc.,  or  SP  at  0  along  OZ, 
together  with  the  couple 

Ayi  +-P»yt +^»y8  +eto.,  or  I,Py,  in  the  plane  yz, 
and  the  couple 

P,a:i  +  Pt^g+P^x,  +etc.,  or  SPx  in  the  plane  xz. 

The  first  couple  tends  to  turn  the  body  from  the  axis  of  y 
to  that  of  z  round  the  axis  of  x,  and  the  second  couple 


8  IN  BPACMa. 

r  a  System  of 
Xigid   Body   in 

be  forces,  and  let 


Plfl<32 

1  also  at  JVj,  two 
»  P,.    Then  the 


»,atJV,; 
1  at  0. 

and  this  conplc 

is  parallel  to  its 

(Art.  62).    The 

the  couple  is  in 

he  whole  system 


ong  OZ, 


the  plane  yt^ 


he  plane  xz. 

am  the  axis  of  y 
second  couple 


MQVTLIBRIPS  Of  PARALLEL  FORVBa  IN  SVACB.     87 

tends  to  turn  the  body  from  the  avis  of  a;  to  that  of  t 
round  the  axis  of  y.  It  is  customary  to  consider  those 
couples  as  positive  which  tend  to  turn  the  body  in  tlie 
direction  indicated  by  the  natural  order  of  the  letters,  i.  «., 
positive  from  x  to  y,  round  the  «-axi8 ;  from  y  to  «  round 
the  z-axis ;  and  from  z  to  x  round  the  y-axis ;  and 
negative  in  the  contrery  direction. 

Hence  the  moment  of  the  first  couple  is  +^Py,  and 
therefore  OX  is  its  axis  (Art.  60) ;  and  the  moment  of 
the  second  couple  is  —I,Px,  and  therefore  OV  \»  its  axis. 
The  resnltant  of  these  two  couples  is  a  single  couple  whose 
axis  is  found  (Art.  66)  by  drawing  OL  (in  the  positive 
direction  of  the  axis  of  x)  —  ^Py,  and  OM  (in  the  nega- 
tive direction  of  the  axis  of  y)  =  ZPx,  and  completing  the 
parallelogram  OLOM.  If  00,  the  diagonal,  is  denoted  by 
0,  we  have 


and 


i2  =  2:P; 


R  being  the  resultant  force. 

Now  since  this  single  force,  R,  and  this  single  couple,  O, 
cannot  produce  equilibrium  (Art  64,  Cor.),  we  must  have 
R  =  0,  and  0  =  0,  and  O  cannot  be  =  0  unless  I,Px  =  0 
and  £Py  =  0 ;  the  conditions  therefore  of  equilibrium  are 


R  =  0, 
IPx  =  0,     SPy 


0. 


Hence,  the  conditions  of  equilibrium  of  parallel  forces  in 
space  are: 

7%0  sum  of  the  forces  must  =  0. 

I%e  sum  of  the  moments  of  the  forces  mth  respect  to 
every  plane  parallel  to  them  must  =  0. 


wm 


88 


SqUrLtBRWM  OF  FORCXa. 


i 


KM. 


Fig.  33 


65.  ConditioiiB  of  EqtdUbriiim  of  a  System  of 
Forces  acting  in  any  Direction  on  a  Rigid  Body  Sn 
Space. — Let  P,,  P,,  Pj,  etc.,  denote  the  forces,  and  let 
them  be  referred  to  three  rectangular  axes,  OX,  OY,  02; 
let  (a;,,  yi»  «i),  (*i»  y».  2|).  etc.,  be  the  pointa  of  applica- 
tion of  P,,  Pf,  etc. 

Let  J,  be  the  point  of  application  of 
Pjj  resolre  Pj  into  components  X,, 
J",,  Zj,  parallel  to  the  co-ordinate  axes. 
Lot  the  direction  of  Z^  meet  the  plane 
xy  at  if  J,  and  dr»»w  if,iVi  perpendicu-  ^ 
lar  to  OX  Apply  at  iV\  and  also  at  0 
two  opposing  forces  each  equal  aud  par- 
allel to  Z,.  Hence  Z^  at  J,  or  Jf,  io  equivalent  to  Z,  at 
O,  and  two  couples  of  which  the  former  has  its  moment  = 
Z,  X  iV",iIf,  r=  Z,yj,  and  may  be  supposed  to  act  in  tho 
plane  yz,  and  the  latter  has  its  moment  =  Z,  x  ONi  = 
—  Z^x^  aud  act*  in  the  plane  zar. 

Hence  Z,  is  replaced  by  Z,  at  0,  i.  couple  Zi^j  in  the 
plane  yz,  and  a  couple  —  Z,a;j  (Art.  64)  in  the  plane  zx. 
Similarly  X,  may  be  replaced  by  X,  at  0,  a  couple  X,«, 
in  the  plane  zx,  and  a  couple  —  X,y,  in  the  plane  xy. 
And  F,  may  be  replaced  by  T,  at  0,  a  couple  JT^a;,  in  the 
plane  xy,  and  a  couple  —  Y^Zi  in  the  piano  yz.  Therefore 
the  force  P,  may  be  replaced  by  X,,  F,,  Z,,  acting  at  6>, 
and  three  couples,  of  which  the  moments  are,  (Art.  54), 

Z,y,  —  Fi«,  in  the  plane  yz,  around  tho  axis  of  a?, 
X,«,  —  ZiX^  in  the  plane  zx,  around  the  axis  of  y, 
I'jo;,  —  X^y,  in  the  plane  xy,  around  the  axis  of  «r. 

By  a  similar  resolution  of  all  the  forces  we  shall  Lave 
them  replaced  by  the  forces 

sx,  sr,   iz, 

acting  at  0  along  the  axes,  and  the  couples 


a  SyBtein  of 
Rigid  Body  ^ 

>e  forcoa,  and  ht 

,  OX,  or,  OZ; 

tointfl  of  applica- 


,f 


4 


*z. 


r^tT' 


!J-X 


ivalent  to  Z,  at 

KB  ita  moment  = 

ed  to  act  in  the 

Z,  X  ON^  = 

iple  Ziff^  in  the 

n  tho  plane  zx. 
a  conple  X,«, 

a  the  plane  xy. 

pie  J",*,  in  the 
yz.  Therefore 
,,  acting  at  0, 

■e,  (Art.  54), 

tho  axis  of  x, 
the  axis  of  y, 
tho  axis  of  z. 

!s  we  shall  Lave 


tiimuMm  ■MMmn)'^ 


BqviLiBntmt  of  forces.  89 

£  (Zy  —  F«)  =  L,  suppose,  in  tho  plane  yt, 
S  (Xz  —  Zx)  =  M,  suppose,  in  the  plane  tz, 
S  (  Fa;  —  Xy)  =  N,  suppose,  in  the  plane  xx. 

Let  U  be  the  resultant  of  the  forces  which  act  at  0;  a, 
b,  c,  the  angles  its  direction  makes  with  the  axes ;  then 
(Art.  38), 

R*  =  (1X)»  +  (2F)»  +  (^Zy» 

sx        ,     sr  xz 

coc  a  =  -jT->    cos  0  —  -g- »    cos  c  =  -pT  • 

Let  G  be  the  moment  of  the  couple  ^"rhich  is  the  result- 
ant of  the  three  couples,  L,M,N;  A,  ^<,  v,  the  angles  its 
axis  makes  with  the  co-ordinate  axes  ;  then  (Art.  5G,  Sch.), 


<]h  =  D  +  M^  +  IP, 


,       L  M 

cos  A  =  -^,    cos  /i  =  -g, 


COS  V  = 


N 


Therefore  any  system  of  forces  acting  in  any  direction  on 
a  rigid  body  in  space  may  always  be  reduced  to  a  single 
force,  R,  and  a  single  couple,  0,  and  cannot  therefore  pro- 
duce equilibrium  (Art  64,  Oor.).  Hence  for  equilibrium 
we  must  have  ^  =  0  and  Q  =  Q;  therefore 

ijLXf  -H  (sr)»  +  {-LZf  =  0, 
and  Z»  -H  iP  +  ^  =  0. 

These  lead  to  the  six  conditions, 

SX=0,    2r=0,    SZ=0, 

£  (%  -  r«)  =  0,     1  (Xz  —  Zx)  =r  0, 

l,{Tx^Xy)  =  0. 


fel- 


90 


KXAMPLXa. 


EXAMPLES. 

1.  If  the  weights,  1,  2,  3,  4,  6  lbs.,  act  pevpendicalarly 
to  a  stniight  line  at  the  reep^ctive  distances  of  1,  2,  3,  4, 

5  feet  from  one  extremity,  find  the  resultant,  and  tho  dis- 
tance of  its  point  of  application  from  the  first  extremity. 

Ans.  i?  =  15  lbs.,  x  —  ^  feet. 

2.  Four  weights  of  4,  —7, 8,  —8  lbs.,  act  perpendicularly 
to  a  straight  line  at  the  points  A,  B,  C,  D,  so  that  AB  = 

6  feet,  BC  =  4  feet,  CD  =  2  feet ;  find  the  resultant  and 
its  point  of  application,  G. 

Ana.  i?  =  2  lbs.,  AG  =  2  feet. 

3.  Two  parallel  forces  of  23  and  42  lbs,,  act  at  the  points 
A  aud  B,  14  inches  apart;  find  GB  to  three  places  of 
decimals.  -Ans.  4.954  ins. 

4.  Two  weights  of  3  cwts.  2  qrs.  15  lbs.,  and  1  cwt,  3  qis. 
25  lbs.  are  su;  ported  at  the  points  A  and  B  of  a  straight 
line,  the  length  AB  -  3  feet  7  inches ;  find  AG  to  tlirfco 
places  of  decimals  offset  Ans.  1.268  ft. 

0.  A  bar  of  iron  15  inches  long,  weighing  12  lbs.,  and  of 
uniform  thickness,  has  a  weight  of  10  lbs.  suspended  from 
one  extremity ;  at  what  point  must  the  bar  be  supported 
that  it  may  jnst  balance. 

Tbe  weight  of  the  bar  acts  at  its  centre. 

An«.  4-^  in.  from  the  weight. 

6.  A  bar  of  uniform  thickness  weighs  10  lbs.,  and  is 
6  feet  long ;  weights  of  rf  lbs.  and  5  lbs.  are  suspended  from 
its  extrumitios ;  on  wlut  point  wiU  it  balance  ? 

Am.  b  in.  from  the  centre  of  the  bar. 

7.  A  beam  30  feof  long  balances  itself  on  a  point  at  one- 
third  of  its  length  fcom  the  thioker  end  ;  but  when  a  weight 
of  10  lbs.  is  suspemtod  from  tbe  smaller  end,  tho  prap  must 


Mte 


pei'pendicalarly 
»8  of  1,  2,  3,  4, 
ut,  and  tho  dis- 
■st  extremity. 
«  =  3|  feet. 

perpendicularly 
),  80  that  AB  = 
e  resultant  and 

A.G  =  2  feet. 

ict  at-  the  points 
three  places  of 
ru.  4.954  ins. 

nd  1  cwt.  3  qid. 
B  of  a  straight 
id  AG  to  thrfco 
Ins.  1.268  ft. 

12  lbs.,  and  of 
suspended  from 
ix  be  supported 


n  the  weight. 

10  lbs.,  and  is 
mspendod  from 

0? 

re  of  tho  bar. 

a  point  at  one- 
when  a  weight 
tho  prap  must 


n 


XIXAMPLSa. 


n 


be  moved  two  feet  towards  it,  in  order  to  maintaiu  the 
equilibrium.     Find  tho  weight  of  the  beam.  Am.  90  lbs. 

8.  A  uniform  bar,  4  feet  long,  weighs  10  lbs.,  and  weights 
oi  30  lbs.  and  40  lbs.  are  appended  to  its  two  extremities ; 
where  must  the  flilcrum*  be  placed  to  produce  equilibrium  P 

An».  8  in.  from  the  centre  of  the  bar. 

9.  A  bar  of  iron,  of  uniform  thickness,  10  ft  long,  and 
weighing  1^  cwt,  is  supported  at  its  extremities  in  a  hori- 
zontal position,  and  carries  a  weight  of  4  cwt  suspended 
from  a  point  distant  3  ft  from  one  ex^^^mity.  Find  the 
pressures  on  the  points  of  support 

^ns.  3.55  cwt,  and  1.05  owt 

l'^  \  bar,  each  foot  in  length  of  which  weighs  7  lbs., 
rests  upon  a  fulcrum  distjint  3  feet  ttoTa  one  extremity ; 
whttt  must  be  its  length,  that  a  weight  of  71^  'Hs.  sus- 
pended from  ihat  extremity  may  just  be  balai  i?ed  by 
20  lbs.  suspended  from  the  other  r  Ans.  9  ft 

11.  Five  equal  parallel  forces  act  at  6  angles  of  a  regular 
hexagon,  whose  diagonal  is  a  ;  find  the  point  of  application 
of  Lheir  resultant 

Ana.  On  the  diagonal  passing  through  the  sixth  angle,  at 
a  distance  from  it  of  \a. 

12.  A  body,  /*,  suspended  from  ona  end  of  a  lever  with- 
out weight,  is  balanced  by  a  weight  of  1  lb.  at  the  other 
end  of  the  ie^  er ;  and  when  tho  fulcrum  is  removed 
through  naif  the  length  of  the  lever  it  requiiea  10  lbs.  to 
balance  P ;  find  the  weight  of  P.      Ana.  5  lbs.  or  2  lbs. 

13.  A  carriage  wheel,  whoM  weigbi  is  W  and  radius  r, 
rests  upon  a  level  road ;  show  tha*;  the  force,  F,  necessary 
to  draw  the  wheel  ever  an  obstacle,  of  height  A,  is 

„V2rA-A» 


F=z  W 


-T" 


*  nto  ■wpo't  oil  whieh  U  N^s. 


0S 


BXAMPLEa. 


14.  A  beam  of  nniform  thickoesB,  5  feet  long,  weighing 
10  lbs.,  ia  Bupported  on  two  props  at  the  ends  of  the  beura ; 
iind  where  a  weight  of  30  lbs.  must  be  placed,  so  that  the 
pressures  on  the  two  props  may  be  15  lbs.  aud  ib  lbs. 

Ana,  10  ins.  from  the  centre. 

15.  Forces  of  3,  4,  5,  0  lbs.  act  at  distances  of  3  ins., 
4  ins.,  5  ii  s.  6  ins.,  from  the  end  of  a  rod  ;  at  what  distance 
from  the  same  end  does  the  resultant  act  ? 

Ans.  4|  inches. 

16.  Four  vertical  forces  of  4,  6,  7,  9  lbs.  act  at  the  four 
corners  of  a  squai'e ;  find  the  point  of  application  of  the 
resultant.    Ana.  -f^oi  middle  line  from  one  of  the  sides. 

17.  A  flat  board  12  ins.  square  is  suspended  in  a  hori- 
zontal position  by  strings  attached  to  its  four  comers,  A, 
B,  C,  D,  and  a  weight  equal  to  the  weight  of  the  board  is 
Inid  upon  it  at  a  point  3  ins.  distant  from  the  side  AB  and 
4  ins.  from  AD ;  find  the  relative  tensions  in  the  four 
strings.  An$.  As  | :  | :  | :  ^. 

18.  A  rod,  AB,  moves  freely  about  the  end,  B,  as  on  a 

hinge.     Its  weight,  W,  acts  at  its  middle  point,  and  it  is 

kept  horizontal  by  a  string,  AG,  that  makes  an  angle  of  45" 

with  t.     Fina  the  tension  in  the  stiinr.  .         W 

"  Ana.  — — • 


19.  A  rod  10  inches  long  can  turn  freely  about  one  of 
its  ends  ;  a  weight  of  4  lbs.  is  slung  to  a  point  3  ins.  from 
this  end^  and  the  rod  is  held  by  a  string  attached  to  its  free 
end  and  inclined  to  it  at  an  angle  uf  120° ;  find  the 
tension  in  the  striug  when  the  rod  is  horizontal. 

Ana.  \  v/S  lbs. 

20.  Two  forces  of  3  lbs.  and  4  lbs.  act  at  the  extremities 
cf  a  straight  lever  12  ins.  long,  and  inclined  to  it  at  angles 
of  120°  aud  136°  respectively ;  find  the  position  of  the 
fulcrum.         j^na.  (8-3  V6)  x  9.6  ins.  from  one  end. 


Ji 


long,  weighing 
i  of  the  beuni ; 
id,  80  tliat  the 
J  25  lbs. 
1  the  (cntre. 

ices  of  3  ins., 
:  what  distance 

>.  4|  inches. 

ct  at  the  foar 
lication  of  the 
of  the  sides. 

ied  in  a  hori- 

ur  corners,  A, 

f  the  board  is 

side  AB  and 

I  in  the  four 

id,  B,  as  on  a 
)int,  and  it  is 
angle  of  45" 

Ans.  — —  • 
V2 

abont  one  of 
nt  3  ins.  from 
led  to  ita  free 
0° ;  find  the 
al. 
t  \/3  lbs. 

le  extremities 
}  it  at  angles 
sition  of  the 

m  one  end. 


MXAMPLKS.  ^ 

21.  Find  the  true  weight  of  a  body  which  is  found  to 
weigh  8  ozs.  and  9  ozs.  when  placed  in  each  of  the  scale- 
pans  of  a  false  balance.  ^,„.  eVaozs. 

22.  A  beam  3  ft.  long,  the  weight  of  which  is  10  lbs., 
and  acts  at  its  middle  point,  rests  on  a  rail,  with  4  lbs.  hang- 
ing from  one  end  and  13  lbs.  from  the  other ;  tind  the  point 
at  which  the  beam  is  supported ;  and  if  the  weights  at  thu 
two  ends  change  plaoee,  what  weight  meat  be  added  to  thn 
lighter  to  preserve  equilibrium  ? 

Ans.  12  ins.  from  one  end  ;  27  Ibe. 

23.  Two  forces  of  4  lbs.  and  8  lbs.  act  at  the  ends  of  a 
bar  18  ins.  long  and  make  angles  ot  120°  and  00°  with  it; 
find  the  point  in  the  bar  at  which  the  resultant  acts. 

An*.  H  (*  —  VS)  ins.  from  the  4  lbs.  end. 

24.  A  weight  of  24  lbs.  is  suspended  by  two  flexible 
strings,  one  of  which  is  horizontal,  and  the  other  is  inclined 
at  an  angle  of  30°  to  the  vertical.  What  is  the  tension  in 
each  string  ?  ^^s.  8  V3  lbs. ;  16  ^3  lbs. 

25.  A  pole  12  ft  long,  weighing  25  lbs.,  rests  with  one 
end  against  the  foot  of  a  wall,  and  frotn  a  point  2  ft.  from 
the  other  end  a  cord  mns  horizontrtlly  to  a  point  in  the 
wall  8  ft.  from  the  ground  ;  find  the  tension  of  the  cord  and 
the  pressure  of  the  lower  end  of  the  pole. 

Ans.  11.25  lbs.;  27.4  lbs. 

20.  A  body  weighing  6  lbs.  is  placed  on  a  smooth  piano 
which  is  inclined  at  30°  to  the  horizon  ;  find  the  two  direo- 
tionn  in  which  a  force  equal  to  the  body  may  act  to  produce 
equilibrium.  Also  find  vha*.  is  the  pressure  on  the  plant 
in  each  case. 

Atu.  A  force  at  60°  with  the  pUne,  or  vertically  upwards ; 
R  =  ey/3,  or  0. 

27.  A  rod,  AB,  5  ft.  long,  without  weight,  is  hnng  fn>m 
a  point,  0,  by  two  strings  which  are  attached  to  its  ends 


i 


94 


SXAMPLSa. 


and  to  the  point ;  the  atring,  AG,  is  3  ft^  and  BO  is  4  ft  in 
length,  and  a  weight  of  2  lbs.  is  hang  from  A,  and  a  weight 
of  3  lbs.  from  B  ;  find  the  tensions  of  the  strings. 

Ana,  V5lbs.;  2  V5  lbs. 

28.  Find  the  height  of  a  cylinder,  which  can  just  rest  on 
an  inclined  plane,  the  angle  of  which  is  60°,  the  diameter 
of  the  cylinder  being  6  ins.  and  its  weight  acting  at  the 
middle  point  of  its  axis.  Am.  3.46  ins. 

29.  Two  equal  weights,  P,  Q,  are  connected  by  a  string 
which  passes  over  two  smooth  pegs,  A,  B,  situated  ii>  a 
horizontal  line,  and  supports  a  weight,  W,  which  hangs 
from  a  smooth  ring  through  which  the  string  pusses ;  find 
the  position  of  equilibrium. 

Atu.  The  depth  of  the  ring  below  the  line 

W 


AB  = 


2  ^4/^  -  W 


AB. 


80.  The  resultant  of  two  forces,  P,  Q,  acting  at  an  angle, 
$,  is  =  {2m  +  1)  \//*  +  Q^;  when  they  act  at  an  angle, 


a  — 6,  it  is  =  (2f»  —  1)  VP*+  Q*;  show  that  tan  e  = 
« 

tn  —  l 

m  +  1 

81.  A  uniform  heavy  beam,  AB  =  2a, 
rests  on  a  smooth  peg,  P,  and  against  a 
smooth  vertical  wall,  AD ;  the  horizontal 
distance  of  the  peg  from  the  wall  l)cing 
h ;  find  the  inclination,  0,  of  the  beam  to 
the  vertical,  and  the  pressnrcs,  R  and  S,  on  the  wall  and  peg. 

Ans.  e  =  s.n-.  (*)*;  S  =  w(^f;  R  =  ff^^^. 

83.  Two  equal  smooth  cylinders  rest  in  oortaot  on  two 
smooth  planes  inclined  at  angles,  a  and  (i,  to  the  horizon ; 


Pig.M 


■■-  9.mtrsfmmvmt 


1  BO  is  4  ft  in 
i,  and  a  weight 
ings. 

1.;  2  V5  1b8. 

an  just  rest  on 
,  the  diameter 
:  acting  at  the 
ns.  3.46  ins. 

;d  by  a  string 
situated  in  a 
which  hangs 

g  passes;  find 


g  at  an  angle, 
at  an  angle, 

that  tan  6  = 


rig.J4      i 

wall  and  peg. 


A* 

ntaot  on  two 
tlie  horizon; 


MXAMPLHa. 


90 


find  the  inclination,  9,  to  the  horizon  of  the  line  joining 
their  centres.  Ans.  tan  0  =- 1  (cot  a  —  cot  j3). 

33.  A  beam,  6  ft.  long,  weighing  5  lbs.,  rests  on  a  ver- 
tical prop,  CD  =  ^  ft. ;  the  lower  end,  A,  is  on  a  hori- 
zontal plane,  an-l  is  prevented  from  sliding  by  a  string, 
AD  =  3^  f  t. ;  find  the  tension  of  the  string. 

Ans.  T—\  lbs. 

34.  A  uniform  beam,  AB,  is  placed  with  one  end,  A, 
inside  a  smooth  hemispherical  bowl,  with  a  point,  P,  rest- 
ing on  the  edge  of  the  bowL  If  AB  =  3  times  the  radius 
R,  find  AP.  Ans.  AP  =  1.838  R. 

35.  A  body,  weight  W,  is  suspended  by  a  cord,  length  I, 
from  the  point  A,  in  a  horizontal  plane,  and  is  thrust  out 
of  its  vertical  position  by  a  rod  without  weight,  acting  at 
another  point,  B,  in  the  horizontal  plane,  such  that 
AB  =  rf,  and  making  the  angle,  6,  with  the  plane ;  find 


the  tension,  T,  of  the  cord. 


Am.  r  =  IF  ^  cot  <». 


36.  Two  heavy  uniform  bars,  AB  and 
CD,  movable  in  a  vertical  plane  about 
their  extremities.  A,  D,  which  rest  on  a 
horizontal  plane  and  are  prevented  from 
sliding  on  it;  find  their  position  of 
equilibrium  when  leaning  against  each 
other. 

Let  the  bars  rest  against  each  other  at  B,  and  let 
AD  =  o,  AB  =  h,  CD  =  «?,  BD  =  a;,  W  and  }\\  =  the 
weighte  of  AB  and  CD,  respectively  acting  at  thoir  middle 
points ;  then  we  have 

2««  »r  (a»  -h  4»  -  a^)  =  c  FT,  (a«  -H  *•  -  «»)  (i«  +  a^  -  ««), 

which  is  an  equation  of  the  fifth  degree,  and  hence  always 
has  one  real  root,  the  value  of  which  may  be  determined 
when  numbers  are  pat  for  <t,  b,  and  c 


96 


BXAMPLMa. 


87.  A  parabolic  curve  is 
placed  in  a  vertical  plane  with 
itfl  axis  vertical  and  vertex 
downwards,  and  inside  of  it, 
and  against  a  peg  in  the  focas, 
and  against  the  concave  arc,  a 
smooth  uniform  and  heavy 
beam  rests ;  required  the  posi- 
tion of  equilibrium. 

Let  PB  be  the  beam,  of 
length  /,  and  of  weight  W, 
resting  on  the  peg  at  the  focus, 
F ;  let  AF  =:  ;>  and  AFP  =  ft 


♦w 


rrs.M 


Awi.  0  =  3co8-»(|)* 


38.  Find  the  form  of  the  curve  in  a  vertical  plane  such 
that  .1  heavy  bar  resting  ou  its  concave  s'de  and  on  a  peg  at 
a  given  point,  say  the  origin,  may  be  at  rest  in  all 
positions. 

Ans.  r  =  y  +  k  eeo  e,  in  which  I  =  the  length  of  the 
bar,  k  an  arbitrary  constant,  and  d  the  inclination  of  the 
bar  to  the  vertical.  It  is  the  equation  of  the  conchoid  of 
Nicomedes. 

39.  A  rod  whose  centre  of  gravity  is  not  its  middle  point 
is  hung  from  a  smooth  peg  by  means  of  a  string  attached 
to  its  extremities  ;  find  the  position  of  equilibrium. 

A)U.  There  are  two  positions  in  which  the  rod  hanga 
vertically,  and  there  is  a  third  thus  defined  :— Let  F  be  the 
extremity  of  the  rod  remote  {torn  the  centre  of  gravity,  k 
the  distance  of  the  centre  of  gravity  from  the  middle  point 
of  the  rod,  2a  the  length  of  the  string,  and  2c  the  length  of 
the  rod  ;  then  measure  on  the  string  a  length  FP  from  F 

equal  to  o  (l  +  - 1,  and  place  the  point  P  over  the  peg. 

This  will  define  a  third  poutiou  of  equilibrium. 


3  co«r«  (^f. 

ical  plane  such 
jnd  on  a  peg  ut 
it   rest    in    all 


length  of  the 

ination  of  the 

le  conchoid  of 


middle  point 

itring  attached 

brium. 

he  rod  hangs 

-Let  F  be  the 

of  gravity,  k 

middle  point 

the  length  of 

FP  from  F 

)Ter  the  peg. 

a. 


MXAMPLMa. 


W 


40.  A  smooth  hemisphere  is  flxdd  on  a  horizontal  plane, 
with  its  convex  side  turned  upwards  and  its  base  lying  in 
the  plane.  A  heavy  uniform  beam,  AB,  rests  against  the 
hemisphere,  its  extremity  A  being  just  out  of  contact  with 
the  horizontal  plane.  Supposing  that  A  in  attached  to  a 
rope  which,  passing  over  a  smooth  pulley  placed  vertically 
over  the  centre  t^t  the  hemisphere,  sustains  a  weight,  tind 
the  position  of  tqnilibrium  of  the  beam,  and  the  requisite 
magnitude  of  the  snspended  weight 

Ans.  Let  W  be  the  weight  of  the  beam,  2o  its  length,  P 
the  suspended  weight,  r  the  radius  of  the  hemisphere,  h 
the  height  of  the  pulley  above  the  plane,  6  and  ^  the 
inclinations  of  the  beam  and  rope  to  the  horizon ;  then  the 
position  of  equilibrium  is  defined  by  the  equations. 


r  oosec  0  =  A  cot  ^, 
r  cosec*  0  =  a  (tan  ^  -f-  cot  6), 
which  give  the  single  equation  for  6, 

r  (r  —  a  sin  9  cos  6)  =  ah  sin*  B. 


Also 


W 


sin  9 


cos  {p  —  (?) 


=  W 


a  sin»  e  Vr*  +  A»  sin*  (9 


(1) 
(«) 

(8) 

(4) 


41.  If,  in  the  last  example,  the  position  and  magnitude 
of  the  beam  be  given,  find  the  locus  of  the  pulley. 

Ans.  A  right  line  joining  A  to  the  point  of  intersection 
of  the  reaction  of  the  hemisphere  and  W. 

42.  If,  in  the  same  example,  the  extremity.  A,  of  the 
beam  rest  against  the  plane,  state  how  the  nature  of  the 
problem  is  modified,  and  find  the  position  of  equilibrium. 

Ans.  The  suspended  weight  must  be  given,  instead  of 
being  a  result  of  calculation.    Equation  (1)  still  holds,  but 


BXAMPLSa, 


li  ■' 


not  (2) ;  and  tho  position  of  equilibrium  is  defined  by  the 
equation 

Ph*  co8^  0  =  War  Bin»  0. 

43.  If  the  fixed  hemisphere  be  replaced  by  a  fixed  sphere 
or  cylinder  resting  on  the  plane,  and  the  extremity  of  the 
beam  rest  on  the  ground,  find  the  position  of  equilibrium. 

Ans.  If  A  denote  the  vertical  height  of  the  pulley  above 
the  point  of  contact  of  the  sphere  or  cylinder  with  the 
plane,  we  have 

r  cot  5  =  A  cot  ^, 

Pr  (1  +  cot  5  cot  0)  cos  0  =  FTa  cos  0. 

44.  One  end,  A,  of  a  heavy  uniform  beam  rests  against  a 
smooth  horizontal  plane,  and  the  other  end,  B,  rests  against 
a  smooth  inclined  plane  ;  a  rope  attached  to  B  passes  over 
a  smooth  pulley  situated  in  the  inclined  plane,  and  sustains 
a  given  weight;  find  the  position  of  equilibrium. 

Let  6  be  the  inclination  of  the  beam  to  the  horizon,  a  the 
inclination  of  the  inclined  plane,  W  the  weight  of  the  beam, 
and  P  the  suspended  weight ;  then  the  position  of  equili- 
brium is  defined  by  the  equation 

cos  «  ( ITsin  «  -  2P)  =  0.  (1) 

Hence  we  draw  two  conclusions: — 

(o)  If  the  given  quantities  satisfy  the  equation  IF  sin  a 
—  2P  =  0,  the  beam  will  rest  in  all  positions. 

(*)  There  is  one  position  of  equilibrium,  namely,  that  in 
which  the  beam  is  vertical. 

This  position  requires  that  both  planes  be  conceived  as 
prolonged  through  their  line  of  intersection. 

45.  A  uniform  beam,  AB,  movable  in  a  vertical  plane 
about  a  smooth  horizontal  axis  fixed  at  one  extremity,  A,  is 


li 


•taatV  fttatrar^  i::tl, 


defined  by  the 


r  a  fixed  sphere 
tremity  of  the 
t  equilibrium, 
iie  pulley  above 
uder  with  the 


cos  0. 

rests  against  a 
B,  rests  against 
>  B  passes  oyer 
le,  and  sustains 
um. 

horizon,  a  the 

\t  of  the  beam, 

ition  of  equili- 

(1) 
nation  TFsin  « 

IB. 

lamely,  that  in 
)e  conceived  aa 

vertical  plane 
xtremity.  A,  is 


MXAMPLES. 


99 


attached  by  means  of  a  rope  BC,  whose  weight  is  negligible, 
to  a  fixed  point  0  in  the  horizontal  line  through  A,  such 
that  AB  =:  AC;  find  the  presau.^  on  the  axis. 

Am.   If  d  =  <CAB,  TF  —  weight  of   beam,    the   re- 
action is 

iTr|/4  8in«s  +  Bec»* 


2 


2 


CHAPTER  IV. 

CENTRE  OF  GRAVITY*  (CENTRE  OF  MASS). 

66.  Centre  of  Gravity. — Gravity  i»  the  name  given  to 
the  force  of  attraction  which  the  earth  exerts  on  all  bodies ; 
the  effects  of  this  force  are  twofold,  (1)  statical  in  virtuo  of 
which  all  bodies  exert  pressure,  and  (3)  kinetical  in  virtue 
of  which  bodies  if  unsupported,  will  fall  to  the  ground 
(Art.  15).  The  force  of  gravity  varies  slightly  from  pluce 
to  place  on  the  earth's  surface  (Art.  23) ;  but  at  each  phu;o 
it  is  a  force  exerted  upon  every  body  and  upon  every 
particle  of  the  body  in  directions  that  are  normal  to  the 
earth's  surface,  and  which  therefore  converge  towards  the 
earth's  centre ;  bat  as  this  centre  is  very  distant  compared 
with  the  distance  between  the  particles  of  any  body  of 
ordinary  magnitude,  the  convergence  is  so  small  that  the 
lines  in  which  the  force  of  gravity  acts  are  sensibly  parallel. 

The  centre  of  gravity  of  a  body  is  the  point  of  application 
of  the  resultant  of  aU  the  forces  of  gravity  which  act  upon 
every  particle  of  the  body;  and  since  these  forces  are 
practically  parallel,  tlis  problem  of  finding  its  position  may 
be  treated  in  the  same  way  as  that  of  finding  the  centre  of  a 
system  of  parallel  forces  (Arts.  46,  59,  63).  The  centre  of 
gravity  may  also  be  defined  as  the  point  at  which  the  whole 
ireighl  of  a  body  acts.  If  the  body  be  supported  at  this 
point  it  will  rest  in  any  position  whatever. 

The  weight  of  a  bod«  is  the  resuUant  of  all  the  forces  of 
gravity  which  act  upon  every  particle  of  it,  and  is  equal  in 
magnitude  and  directly  opposite  to  the  force  which  mil  just 
support  the  body.     Since  the  centre  of  gravity  is  here 

*  CsUed  also  OntAw  of  Man  and  Cmtrt  ot  tntrUa ;  ud  the  tenn  OtntroU  hM 
Utelf  come  into  qm  .o  destgnate  It. 


iF    MASS). 

name  given  to 
I  ou  all  bodies ; 
5al  in  virtue  of 
stical  in  virtue 
to  the  {^round 
itly  from  pluce 
t  at  each  plii«o 
nd  ujjon  every 
normal  to  t lie 
re  towards  the 
tant  compared 

any  body  of 
small  that  the 
insibly  parallel. 
t  of  application 
which  act  upon 
ese    forces  are 

position  may 
the  centre  of  a 
The  centre  of 
^hich  the  whole 
)ported  at  this 

n  the  forces  of 
nd  is  equal  in 
vhich  mil  just 
rarity  is  here 

e  term  OenlnM  hu 


CMyTRH  OF  aSAVTTT. 


101 


regarded  as  the  centre  of  parallel  forces,  it  is  more  t'mly 
conceived  of  as  the  "cdntre  of  mass;"  yet  m  deference  to 
usage  we  shall  call  the  point  the  ''centre  of  gravity." 

67.  Planes  of  Symmetry.— Asm  of  Symmetry.— If 

a  homogeneous  body  be  symmetrical  with  reference  to  any 
plane,  the  centre  of  gravity  is  in  that  plane. 

If  two  or  more  such  planes  of  symmetry  intersect  in 
one  line,  or  axis  of  symmetry,  the  centre  of  gravity  is  in 
that  axis. 

If  three  or  more  planes  of  symmetry,  intersect  each  other 
in  a  point,  that  point  is  the  centre  of  gravity. 

By  observing  these  principles  of  the  symmetry  of  the 
figure  there  are  many  cases  where  the  centre  of  gravity  is 
known  at  once ;  thus,  the  centre  of  gravity  of  a  straight 
line  is  its  middle  point.  The  centre  of  gravity  of  a  circle 
or  of  its  circumference,  or  of  a  sphere  or  of  its  surface,  is  its 
centre.  The  centre  of  gravity  of  a  parallelogram  or  of  ita 
perimeter  is  the  point  in  which  the  diagonals  intersect. 
The  centre  of  gravity  of  a  cylinder  or  of  its  surface  is  the 
middle  of  its  axis;  and  in  a  similar  manner  we  sh^ 
frequently  conclude  from  the  symmetry  of  the  figure,  that 
the  centre  of  gravity  of  a  body  is  in  a  particular  line  which 
can  be  at  once  determined. 

When  we  speak  of  the  centre  of  gravity  of  a  line,  we 
are  really  considering  a  material  lino  of  the  same  density 
and  thickness  throughout,  whose  section  is  infinitesimal ; 
and  when  we  consider  the  centre  of  gravity  of  any  surface, 
wf  are  really  considering  the  surface  as  a  thin  uniform 
lamina,  the  thickness  of  which,  being  uniform,  can  be 
neglected. 

68.  Body  Stupended  from  a  Voiat— When  a  body  is 
suspended  from  a  point  about  which  it  can  turn  freely,  it 
will  rest  with  its  centre  of  gravity  in  the  vertical  line  pass itig 

•Qugh  the  point  of  suspension.    For,  if  the  point  of  sus- 


1» 


BODY  aVPPORTKD  ON  A  SURPACB. 


peiision  and  the  centre  of  gravity  are  not  in  a  vertical  line, 
the  weight  acting  vertically  downwards  at  the  contro  of 
gravity  and  the  reaction  of  the  point  of  suspensiou  vert'cally 
upwards  form  a  statical  couple  and  hence  there  will  be 
rotation. 

69.  Body  Supported  on  a  Sufkce.—  When  a  body  is 
placed  on  a  surface  it  will  stand  or  fall  according  as  the 
vertical  line  through  the  centre  of  gravity  falls  within  or 
without  the  base.  For  if  it  falls  within  the  base  the  reaction 
of  the  surface  upward  and  the  action  of  the  weight -down- 
ward will  be  in  the  same  vertical  line,  and  so  there  will  be 
equilibrium.  But  if  it  falls  without  the  base  the  reaction 
of  the  surface  upward  and  the  action  of  the  weight  down- 
ward form  a  statical  couple  and  hence  the  body  will  rotate 
and  fall. 

7D.  Different  Kinds  of  Sqnilibrinm.— According  to 

the  proposition  just  proved  (Art.  69)  a  body  ought  to  rest 
upon  a  single  point  without  falling,  provided  that  its  cent^ 
of  gravity  is  placed  in  the  vertical  line  through  the  point 
which  forms  its  base.  And,  in  fact,  a  body  so  situated 
would  be,  mathematically  speaking,  in  a  position  of  equili- 
brium, though  practically  the  equilibrium  would  not  sub- 
sist The  body  would  be  moved  from  its  position  by  the 
least  force,  and  if  left  to  itself  it  would  depart  further  from 
it,  and  never  return  to  that  position  again.  This  kind  of 
equilibrium,  and  that  which  is  practically  possible,  are 
distinguished  by  the  names  of  unstable  and  stable.  Thus 
an  egg  on  either  end  is  in  a  position  of  unstable  equilibrium, 
but  when  resting  on  it«  side  it  is  in  a  position  of  stable 
equilibrium.  The  distinction  may  be  defined  generally  as 
follows : 

When  the  body  is  in  such  a  position  that  if  slightly  dis- 
placed it  tends  to  reftim  to  its  original  position,  the  equili- 
brium is  stable.    When  it  tends  to  move  further  away  from 


cs. 


emitKg  or  oravttt  of  a  trijuiols.        103 


ft  vertical  lino, 

)  tiie  contra  of 

nsiou  vert'cally 

tbore  ivill  be 


When  a  body  is 
wording  as  the 
falls  within  or 
»8e  the  reaction 
I  weight -do wn- 
>  there  will  be 
le  the  reaction 
weight  down- 
ody  will  rotate 

-According  to 

ought  to  rest 

that  its  cent;^ 

i]gh  the  point 

iy  80  situated 

ition  of  equili- 

onld  not  snb- 

)8ition  by  the 

;  further  from 

This  kind  of 

possible,  are 

stable.    Thus 

'e  equilibrJum, 

tion  of  stable 

generally  as 

f  slightly  dis- 
)n,  the  equili- 
er  away  from 


its  orifiutf  J^sition,  its  equilibrium  is  unstable.  When  it 
reinaim  in  its  new  position,  its  equilibrium  i .  neutral.  A 
sphere  or  o^ndrical  roller,  resting  on  a  horizontal  surface, 
{equilibrium.    In  stahk  equilihrium  the  centre 


IS  m 


of  grwntp  occupies  the  lowest  possible  position;  and  in 
unstable  it  occupies  the  highest  position. 
Wc  shall  first  give  a  few  elementary  examples. 

71.  Oiven  the  Centres  of  Qravlty  of  two  Mauea, 
Ml  and  M„  to  find  the  Centre  of  Qravity  of  the  two 
Masses  as  one  System.— Let  gi,  denote  the  centre  of 
gravity  of  the  mass  if,,  and  g,  the  centre  of  gravity  of  the 
mass  Mg.    Join  g^  g,  and  divide  it  at  the  point,  O,  so  that 

Og.-  M\' 

ma6ses  as  one  system  (Art  46). 


then  G  is  the  centre  of  gravity  of  the  two 


72.  Oiven  tbe  Centre  of  Gravily  of  a  Body  of 
Mass,  M,  and  also  the  Centre  of  Ghnvity  of  a  part 
of  the  Body  of  Mass,  m,  to  find  the  Centre  of 
Qravity  of  the  remainder. — Let  0  denote  the  centre  of 
gravity  of  the  mass,  M,  and  jr,  the  centre  of  gravity  of  the 
mass,  nt|.  Join  Og^  and  produce  it  through  O  tog,,  so  tliat 

^^  =r  ^_* — ,  then  g,  is  the  centre  of  gravity  of  the 

remainder  (Art  45). 

73.  Centre  of  Gravity  of  a  Triangnlar  Figure  of 
Uniform  Thickness  and  Density. — Let  ABO  be  the 

triangle;  bisect  BC  in  D,  and  joiu  AD; 

draw  any  line  bdc  parallel  to  BC  ;  then  it 

is  evident  that  this  line  will  be  bisected  by 

AD  in  d,  and  will  therefore  have  its  centre 

of  gravity  at  d ;  similarly  every  line  in  the 

triangle  parallel  to  BO  will  have  its  centre 

of  gn^vity  in  AD,  and  therefore  the  centre  of  gravity  of  the 

triangle  must  be  somewhere  in  AD. 


Frg.37 


■  ?>*/■' 


104 


CXXTBB   OF  ORAVnr  or  il    TRZAJteLM. 


In  like  )>  .annor  the  pentre  of  gravity  must  lie  on  the  line 
BE  which  joiuB  B  to  the  middle  point  of  AC.  It  is  there- 
fore at  t  le  intersection,  G,  of  AD  and  BE. 

Join  DE,  whih  wiU  be  parallel  to  Afi ;  then  the  triangle^ 
ABO,  DEG',  are  similar :  therefore 


AG 
GD 


AB_  BO 
DE~DO 


i' 


or  GD  =  iAG  =  ^AD. 

Hence,  to  find  ih«  centre  of  gravity  of  a  triangle,  bisect  any  . 
side,  join  the  point  of  bisection  with  the  opposite  angle,  the 
centre  of  gravity  Ivss  one  third  the  way  up  this  bisection. 

Cor.  1.— If  three  equal  particles  be  placed  at  the  vertices 
of  the  triangle  ABC  their  centre  of  gravity  will  coincide 
wich  that  of  the  triangle. 

For,  the  centre  of  gravity  of  the  two  equal  particles  at  B 
and  0  is  the  middle  point  of  BO,  and  the  centre  of  gravity 
of  the  three  lies  on  the  line  joininjj  this  point  to  A. 
Similarly,  it  lief>  on  the  line  joining  B  to  the  middle  of  AC. 
Therefore,  etc. 

Cob.  2.— The  centre  of  gravity  of  any  plane  polygon  may 
be  found  by  dividing  it  into  triangles,  finding  the  centre  of 
gravity  of  each  triangle,  and  then  by  Art  69  deducing  the 
cenb'c  of  gravity  of  the  whole  figure. 

Let  the  coK)rdinate8  of  A,  referred  to  any  axes, 
«.  ;  those  of  B!,  x,,  y„  «, ;  and  those  of  0,  ar„ 


*t » 


Cob.  8. 

be  a;,,  yj,    ..  , 

y„  «, ;  then  (Art  69),  the  co-ordinates,  5,  y,  t,  of  the  centre 
of  gravity  of  three  equal  particles  placed  at  A  B,  0,  respec- 
tively, are 


•  = 


3 


*  ~  3 


+  y. 


;  =  'jL±_«ijtii; 


M.. 


i«ta 


AjreLK. 

t  lie  on  the  line 
0.  It  is  there- 
en  the  trianglcSy 


%ngle,  Usect  any . 
poaite  angle,  the 
'tis  bisection. 

d  at  the  Tcrtices 
ty  will  coincide 

1  particles  at  B 

entre  of  gravity 

point  to  A. 

middle  of  AC. 


18 


ms  polygon  may 
ig  the  centre  of 
9  dedacing  the 


>ed  to  any  axes, 


i> 


those  of  C,  X 

•,  of  the  centre 

A.  B,  0,  respec- 


+  y. 


CJWVTBi   OF  ORAVTTr  OF  A  PTRAXID, 


which  aro  also  the  co-ordinates  of  the<»ntre  of  gravity  «f 
the  triangle  ABO  (Oor.  1). 

74.  Cemtre  of  Qncvity  of  a  Trlangnlar  Pyramid  of 
Unifonn  Density.— Let  D-ABO  be  a  triangular  pyramid; 
bisect  AO  at  E;  join  BE,  DE;  take  EP 
=  ^EB,  tLan  P  is  the  centre  of  gravity  of 
ABC  (Art  73).  Join  FD ;  draw  ab,  h;-,  ea 
parallel  to  AB,  BC,  CA  respectively,  and 
let  DF  meet  the  plane,  abc,  at  /;  join  bf 
and  produce  it  to  meet  DE  at  e.    Then  ^^ 

since  in  the  triangle  ADC,  ac  is  parallel  ^*''*  ° 

to  AO,  and  DE  bisects  AC,  e  ip  tne  middle  point  of  «c;  also 


but 
'herefore 


K-W  -  sL. 
BF  ~  DF  ~  EF  • 

EF  =  IBF, 


therefore /is  the  centre  of  gravity  of  the  triangle  a  Jc  (Ari 
73).  Now  if  we  suppope  the  pyrnmid  to  be  divided  by 
planes  parallel  to  ABC  into  an  indefinitely  great  number  of 
triangular  laminae,  each  of  these  laminae  has  its  centre  of 
fjravity  in  DF.  Henoe  the  centre  of  gravity  of  the  pyramid 
if^  in  DF. 

Again,  take  EH  r=  JED ;  join  HB  cutting  DF  at  G. 
Then,  as  before  the  centre  of  the  pyramid  must  be  on  BH. 
It  is  therefore  at  the  intersection,  G,  of  the  liues  DF 
and  BH. 

Join  FH ;  then  PH  is  parallel  to  DB.  Also,  EP  =  pB, 
therefore  FH  =  JDB ;  and  in  the  similar  triangles,  FGH 
and  BGD,  we  nave 


therefore 


PG  _  FH  _  1 
M  ~  D  B  "~  8 ' 

FG  =  JDG  =  iDF. 


106 


CMSTRM  OF  QRA.  VrrT  OF  A    CONS. 


\.s' 


3 

u 

m 

Eli' 

I 


nonce,  the  centre  of  gravity  of  the  pyramid  ia  one-fourth 
of  the  way  up  ike  line  joining  the  centre  of  gravity  of  the 
base  with  the  vertex.  (Todhuni  a  Statics,  p.  108.  Also 
Pratt's  Mechanics,  p.  63.) 

Cor.  1.— The  centre  of  gravity  of  four  equal  particles 
placed  at  the  vortjices  oi  the  pyramid  coincideo  with  the 
centre  of  gravity  of  the  pyramid. 

Cob.  2.— Let  (x,,  yj,  «i)  be  one  of  the  vertices ;  («„  y„  «,) 
a  second  vertex,  and  so  on ;  let  (i,  y,  i)  bto  the  centre  of 
gravity  of  the  pyramid  ;  then  (Art  59) 

•  =  J  (*,  +  «,  +  ;r,  +  «4).  • 

»  =  i  (yi  +  yi  +  y»  +  ^t). 
;  =  J  («,  +  «,+«,  +  «4). 

Cor.  3.— The  perpendicular  distance  oi'  t'le  centre  of 
gravity  of  a  triangular  pyramid  from  the  base  is  64031  to  { 
of  the  height  of  the  pyramid. 

75.  0«ntn  of  Qnvitj  cX  a  Cone  of  Uniform 
Denaity  havjog  mnf  Plana  Baaa.— Consider  a  pyramid 
whose  base  ij  a  polygon  of  any  number  of  sides.  Divide 
the  base  into  triangles ;  join  the  vertex  of  the  pyramid  with 
the  vertices  of  all  the  triangles ;  ihen  we  may  consider  the 
pyramid  as  composed  of  a  number  of  triangular  pyramids. 
Now  the  centre  of  gravity  of  each  of  these  triangular 
jiyramids  lies  in  a  plane  whose  distance  fr&m  the  base  is 
one-fourth  of  the  height  of  the  pyramid  (Art  74,  Cor.  3) ; 
tf  ,refore  the  centre  of  gravity  of  the  whole  pyramid  lies  in 
this  plane,  ». u,  i*«  perpendicular  distance  from  the  base  is 
one-fourth  of  the  height  of  the  pyramid. 

Again,  if  we  suppose  the  pyramid  to  be  divided  into  an 
i?»deflnitely  great  yiumber  of  laminu.  as  in  Art  74,  each  of 
these  lamina!)  has  its  centre  of  gravity  on  the  right  UiM 


nid  is  om'fourih 
of  gravity  of  the 
:8,  p.  108.    Also 

r  equal  particles 
incidef)  with  the 


tice«;(a;„y„«,) 
to  the  centrt)  of 


)i'  the  centre  of 
Nisc  is  e4a3l  to  \ 

M  of  Uniform 

laidcr  a  pyramid 

of  sides.    Divide 

he  pyramid  with 

nay  consider  tho 

ovular  pyramidfi. 

Iicse   triangnlar 

f^m  tho  base  is 

Art.  74,  Cor.  3) ; 

pyramid  lies  io 

from  the  base  is 

divided  iato  an 
Art  74,  each  of 
the  right  liiM 


csNTRs  OF  Qturmr. 


107 


joining  the  vertex  to  the  centre  of  gNTity  of  the  base ;  and 
hence  the  centre  of  gravity  of  the  whole  pyramid  lies  on 
tills  Une,  and  hence  it  mast  bo  one-fourth  tiie  way  up  this 
line.  There  is  no  limit  to  the  number  of  sides  of  the  poly- 
gun  which  forms  the  base  of  the  pyramid,  and  henoe  they 
may  form  a  continnous  curve. 

Therefoie,  the  centre  of  gravity  of  a  cone  whose  base  is 
any  plane  curve  whatever  is  found  by  joining  the  centre  of 
gravity  of  the  base  to  the  vertex,  and  taking  a  point  one- 
fourth  of  the  tvay  up  this  line. 

76.  Contre  of  Ckavity  of  tho  Fnutom  of  a  Pyra- 

mid. — Let  ALC-abc  (Fig.  38)  be  the  frustum,  formed  by 
ilie  removal  of  the  pyramid,  D-abc,  from  the  whole  pyramid, 
D-ABO ;  let  A,  and  ff  be  the  perpendicalar  heights  of  these 
])yramids,  respectively;  let  m  and  if  denote  their  masses; 
and  let  Xi,  «,,  0  denote  the  prpendicular  distances  of  the 
centres  of  gravity  of  the  pyramids  D-ABO,  and  D-abe,  and 
tho  frustum,  from  the  base  \  then  we  have  (Art  60,  Soh.  1) 


Jf«j  =  i  ( J/"  —  m)  +  ««, ; 


or 


Rat 


mz 


M-m 


».  =  -r; 


i. 


(1) 


4' 


=  (ir-»,)  +  V  =  ^-**»- 


Also,  the  maasea  of  the  pyramids  are  to  each  other  an  the!. 
M)lumos*  by  (1)  of  Art  \%,  and  therefore  as  the  oubsa  of 
tlieir  heights.    Henoe  (1)  becomes 


*  irtiM  bodiM  Mw  iMMMRenmut,  tlw  roiniiiM  or  Ike  weigtato  m«  proportkwal  to 
I  111-  oMwaM,  and  m»f  be  mibi>tlwia4  kn  tlHM. 


108 


SXAMPLSa. 


H- 


H*  +  iTA,  +  A,  * 


(2) 


Instead  of  the  heights  we  may  use  any  two  corresponding 
lines  in  the  lower  and  npper  baaes,  to  which  the  heights  are 
proportional,  as  for  example  AB  and  <dt.  Denoting  these 
lines  by  a  and  h,  and  the  altitude  of  the  firostum  by  h,  {i) 
becomes 

This  is  true  of  a  fnistum  of  a  pynunid  on  any  base,  a 
and  b  being  homologons  sides  of  the  two  ends,  and  hence  it 
is  true  of  the  frustum  of  a  cone  standing  on  any  plane  baae. 


EXAM  PL.ES. 

1.  Find  the  centre  of  gravity  of  a  t/apeioid  in  terms  of 
the  lengths  of  the  two  parallel  sides,  a  and  b,  and  of  the 
line.  A,  joining  their  middle  points. 

Take  moments  with  raferanoe  to  the  longer  pnimllol  tide. 

An$.  On  the  line  bisecting  the  parallel  sides  and  at  a 

A    a  +  2i 


distance  fh>m  its  lower  end  = 


3     a-\'b 


2.  If  out  of  any  cone  a  similar  cone  ic  cut  so  that  their 
axes  are  in  the  same  line  and  their  bases  in  the  same  plane. 
And  the  height  of  the  centre  of  gravity  of  tha  remainder 
above  the  base. 

Take  momenta  with  refefenoe  to  the 


VLV 


+JV 


(2) 


'0  oorresponding 

I  the  heights  are 

Denoting  these 

ostum  hy  A,  (2) 

(3) 

on  any  base,  a 
ds,  and  hence  it 
I  any  plane  base. 


oid  in  t«rm8  of 
d  b,  and  of  the 

lol  tide. 

sides  and  at  a 


t  so  that  their 

the  same  plane, 

thd  remainder 


llfTSOBATIOJf  FORMULA. 


109 


Ana,  J  .  TTZTTt*  "^^^^  *'  "  *^®  height  of  the"^)^iginal 
cone,  and  h',  the  height  of  that  which  is  cut  out  of  it. 

3.  If  out  of  any  cone  another  cone  is  cut  having  the 
same  base  and  their  axes  in  the  same  line,  find  the  height 
of  the  centre  of  gravity  of  the  remain«ier  above  the  base. 

Ans.  J(A  +  Ai),  where  h  and  A,  are  the  respective 
heights  of  the  original  cone  and  the  one  that  is  cut  out 
uf  it. 

4.  If  out  of  any  right  cylinder  a  cone  is  cut  of  the  same 
base  and  height,  find  the  centre  of  gravity  of  the  remainder. 

Ans.  Iths  of  the  height  above  the  base. 

77.   iDvestigKtions    Iiivolvivf   Integration.— The 

general  formula  for  the  co-ordinates  of  the  centre  of  gravity 
vary  according  as  we  consider  a  material  line,  an  area  or 
thin  lamina,  or  a  solid  ;  and  assume  different  forms  accord- 
ing to  the  manner  in  which  the  matter  is  supposed  to  be 
divided  into  mfinitesimal  elements. 

In  either  case  the  principle  is  the  same ;  the  quantity  of 
matter  is  divided  irto  an  infinite  number  of  inficitesimal 
elements,  the  mass  of  the  element  being  dm ;  multiplying 
the  element  by  its  co-ordinate,  x,  for  example,  we  get 
X  '  dm,  which  is  the  moment  of  the  element*  with  respect  to 
the  plane  yt  (Art.  63) ;  and  /x  •  dm  is  the  sum  of  the 
moments  of  all  the  elements  with  respect  to  the  plane  yx, 
and  which  corresponds  to  SPz  of  Art  63.  Also,  /dm  is 
the  sum  of  the  nuases  of  all  the  elements  which  correspond 
to  £P  of  the  come  Article.  Hence,  dividing  the  former  by 
the  latter  we  have 


*  The  momsnt  of  th«  ft>TC*  Mtlng  on  elemAnt  dm  U  Mriotlj  dm  -gic,  bnt  cinM 
the  eooittnt  g  appean  In  botb  terma  of  expreeilnn  ror  co-ordinate*  of  centre  of 
Kravlty,  it  may  be  omitted  and  It  becomet  more  conrenlent  to  ipaak  of  the  ptomtnt 
ot  the  timtmt,  mtaniag  by  It  the  product  of  the  maM  of  the  al«awF>t  dm,  and  it« 
arni.z.  Tha  momant  of  an  atoaaiit  maaanwa  lt«  «g»  a  dateraUUng  the  poaltfcm 
of  the  oantrt  of  gmTUj. 


110 


CKIfTKS  OF  QBA  VITT  OF  A  LiytU 


m 


fx  •  dm 
/dm 


Similarly 


i  =  ^ 


dm 


t  = 


/dm  * 

/t'dm  ^ 
~/dm'* 


(1) 


(2) 


(3) 


the  limits  of  integration  being  determined  by  the  form  of 
the  body  ;  the  sign,  /,  is  used  as  a  general  symbol  of  sum- 
mation, to  be  replaced  by  the  symbols  of  single,  doable,  or 
triple  integration,  according  as  dm  denotes  the  mass  of  an 
elementary  length  or  surface  or  solid.  Hence,  the  co-or- 
dinate of  the  centre  of  gravity  referred  to  any  plane  is  i  tal 
to  the  gum  of  the  moments  of  the  elements  of  the  mass 
referred  to  the  same  plane  divided  by  the  sum  of  the  elements, 
or  the  whole  mass.  If  the  body  has  a  plane  of  symmetry 
(Art.  67),  we  may  take  it  to  be  the  plane  xy,  and  only  (1) 
and  (2)  are  necessary.  If  it  has  an  axis  of  symmetry  we 
may  '^ke  it  to  bo  the  axis  of  z,  and  only  (1)  is  necessary. 

79.  Centn  of  Qn^ity  of  the  Are  of  a  Conro.— If 

the  body  whose  centre  of  gravity  we  want  is  a  material  line 
in  the  form  of  the  arc  of  any  curve,  dm  denotes  the  mass  of 
an  elementary  length  of  the  carve. 

Let  ds  =■  the  length  of  an  element  of  the  curve ;  let 
h  =  the  area  of  a  normal  section  of  the  curve  at  the  point 
ix,  y,  z),  and  let  p  =  the  density  of  the  matter  at  this 
point  Then  (Art.  11),  we  have  dm  =  kpds,  which  is  the 
mass  of  the  element ;  multiplying  this  mass  by  its  co-or- 
dinate, X,  for  example,  we  have  the  moment  of  the  element, 
(kftxds),  with  respect  to  the  plane,  ?  ». 

Hence,  substituting  for  dm  in  (1),  (2),  (8),  of  Art.  77, 
the  linear  element,  kpds,  we  obtain,  for  the  position  of  the 
centre  of  gravity  of  a  b«jdy  in  the  form  of  any  curve,  the 
oquutioni 


u, 

(2) 

(3) 

1  by  the  form  of 
I  symbol  of  Bum- 
single,  doable,  or 
I  the  mass  of  an 
Hence,  the  co-or- 
\ny  plane  is  t  lal 
ents  of  the  mass 
m  of  the  eUmenis, 
Hie  of  symmetry 
ty,  and  only  (1) 
of  symmetry  we 
)  is  necessary. 

9f  aCnnr*.— If 

s  a  material  line 
lotea  the  mass  of 

'  the  curve;  let 
rte  at  the  point 

matter  at  this 
ds,  which  is  the 
ass  by  its  co-or- 

of  the  element, 

(8),  of  Art.  77, 
position  of  the 
any  cnrre,  the 


MXAkPLJSH. 

fkpxdi 


C  := 


-_/kpyds 
^  -  J'kpds  ' 


•  = 


/kptd$ 
fkpds' 


(1) 
(8) 
(8) 


The  quantities  h  and  p  mast  be  given  as  functions  of  the 

position  of  the  point  {x,  y,  «)  before  the  integrations  can 

be  performed. 

If  the  curve  is  of  doable  corvature  all  three  equations 

required.    If  it  is  a  plane  curve,  we  may  take  it  to  be 


are 

in  the  plane  xy,  and  (1)  and  (2)  are  sufficient  to  determine 
the  centre  of  gravity,  since  i  =  0.  If  the  curve  has  an  axis 
of  symmetry,  the  axis  of  *  may  be  made  to  coincide  with 
it,  and  (1)  is  sufficient 

« 

KXAMPL.ES. 

1.  T   find  the  centre  of  gravity  of  a  circular  4ro  of  uni- 
form thickness  and  density. 

Let  BC  be  the  arc,  A  its  middle  point, 
and  0  the  centre  of  the  circle.  Theti  as 
the  arc  is  symmetrical  with  respect  to  OA 
its  centre  of  gravity  must  lie  on  this  line. 
Take  the  origin  at  0,  and  OA  as  axis  of  x. 
Then,  since  k  and  p  are  constant,  (1)  be- 
comes 


fxds 


m 


X  being  the  co-ordinate  of  any  point,  P,  in  the  arc.     Let  9 
be  the  angle  POA,  and  a  the  radius  of  the  circle,  and  let 
=:  the  angle  BOA.    Then 


II 


m 


113 


and 


Hence  m  = 


/!- 


BXAMPLSa. 

2  =  a  COS  9, 
da  =  ade. 

J  COB  Od$ 


COB  e  dO 


=  a 


f*a  do  PdO 


=  a 


Bin  a 


Therefore,  the  distance  of  the  centre  of  gravity  of  the  arc  of 
a  circle  from  tJte  centre  is  the  product  of  the  radius  and  the 
chord  of  the  arc  divided  by  the  length  of  the  arc. 

Cob.— The  distance  of  the  centre  of  gravity  of  a  semi- 
circle* from  the  centre  is 


2a 


3.  Find  tb«  centre  of  gravity  of  the  quadrant,  AT/,  (Fig. 
89),  referred  to  the  co-ordinate  axes  OZ,  OF. 
The  equation  of  the  circle  is 

a*  +  y«  =  a». 
.     ^ <?y  _  Vda?  +  dt^  _ds  _ 

J        axdxy 
.'.,  xde  =z / 

y 


and 


yds  sz  adx, 

J        ache 
ds  = ; 

y 


/ 


which  in  (1)  and  (3),  after  canceling  h  and  p,  give 


rxdx  r  -la 


rdx 


h-sl 


f;») 


=  a 


Bin  a 


\y  of  the  are  of 
radius  and  the 

\rc. 

ivity  of  a  semi- 


tint,  AT/,  (Fig. 
r 


th 


give 


8a 


iza\ 


SXAMPLBS. 


/**      H 


y  = 


r__dx_ 


-v/o»-a? 


hil 


2a 


3.  Find  the  centre  of  gravity  of  the  arc  of  a  cycloid. 
Take  the  origin  at  the  starting  point  of  the  cycloid,  and 
let  the  base  be  taken  as  the  axis  of  x.    The  eqaaUou  of 


the  curve  is 


X  =  a  vers-»  ^  —  (2ay  —  j/*)*  j 

&!  _        dy       __     ds    , 
y*~(2a-y)*~(2a)** 

it  is  evident  that  the  cenire  of  gravity  will  be  in  the  axis  of 
the  cycloid ;  therefore  i=:na;  and  as  k  and  p  are  constant, 
(2)  becomes 

l^_ffdj/_ 
c/q    (2a-y)*_ 

•^«    (2a -y)* 
OoB. — For  the  aio  of  a  semi-cycloid,  we  get 
5  =  fi,    jr  =  |a. 

4.  Find  tbe  centre  of  gravity  of  a  circular  arc  of  uniform 
section,  the  density  varying  as  the  length  of  the  arc  from 
one  extremity. 

Let  AB  (Fig.  39),  be  the  arc ;  let  ft  be  the  density  at  the 
units  distance  from  A,  then  fts  will  be  the  density  at  the 
distance  «  from  A ;  let  OA  be  the  axis  of  x,  and  a  the 
Z.  AOB.  Then,  putting  fu  for  p,  and  a  cos  9,  a  sin  0,  a  (J9, 
and  ofl,  for  x,  y,  dt,  and  a,  in  (1)  and  (2), 


wm 


0  Bin  edB 


0 


Odd 


of  a  loop  of  a 
W,  I  being  the 


to. 


=  o» 


2*-l 


rod,  the  den- 
hc  distance  of 

to  cdndde  with 


CSlfTXE  OF  OMA  VTTT  OF  AN  ARSA. 


116 


7.  Find  the  oentro  of  gravity  of  the  arc  of  a  semi-car- 
dioid,  its  cqaation  being 

r  =  o  (1  +  cos  fl)' 

Am.  The  co-ordinates  of  the  centre  of  gravity  referred 
to  the  aiis  of  the  curve  and  a  perpendicalar  mrongh  the 
cnsp,  as  axes  of  x  and  y,  are 


^. 


'i^ 


«  = 


y  =  ^«- 


c 


^ 


79.  Centra  of  Qravity  of  a  Plane 
Area. — Let  ABCD  be  an  area  bounded 
by  the  ordinates,  AC  and  BD,  the  carve 
AB  whose  equation  is  given,  and  the  axis 
of  a; ;  it  is  required  to  find  the  centre  of 
gravity  of  this  area,  the  lamina  (Art  67) 
being  supposed  of  uniform  thickness  and  density.  We 
divide  the  a.ea  into  an  infinite  number  of  infinitesimal 
elements  (Art.  77).  Suppose  this  to  be  done  by  drawing 
ordinates  to  the  curve.  Let  PM  and  QN  be  tw6  consecu- 
tive ordinates,  let  (x,  y)  be  the  point,  P,  and  let  g  be  the 
centre  of  gravity  of  the  trapezoid,  MPQN,  whose  breadth  is 
dx  and  whose  parallel  sides  are  y  and  y  +  dy.  The  area  of 
this  trapezoid  \Aydx,  (CaL,  Art  184). 

Let  p  be  the  density  and  k  the  thickness  of  the  lamina. 
Then  (Art  11)  we  have  dm  —  kpy  dx,  which  is  the  mass 
of  the  element  MPQlJ ;  multiplying  this  mass  by  its  co-or- 
dinate, X,  for  example,  we  have  tuo  moment  of  the  element 
{lepxydx),  with  respect  to  OY,  and  multiplying  by  the 
other  co-ordinate,  \y,  we  have  the  moment  with  respect  to 
OX.  Hence,  substituting  for  dm  in  (1)  and  (2)  of  Art.  77, 
the  surface  element,  hpy  dz,  and  remembering  that  k  and  p 
are  constants,  we  obtain,  for  the  position  of  the  centre  of 
gravity  of  a  body  in  the  form  of  a  plane  area,  the  equations, 


116  MXA.MPLX& 

yydx.'  *-*jf^*  (1) 

the  integrations  extending  over  the  whole  area  CABD. 

EXAM  Pt.ES. 

1.  Find  the  centre  of  gravity  of  the  area  of  a  semi-parab- 
ola whose  equation  is  y*  r=  2px. 

Let  a  =  the  axis,  and  *  the  extreme  ordinate,  then  we 
have  from  (1) 

J^V2p  z*  dx       J*^  dx 


—  _L 


_  *'o 


J  V^x^dx       Jx^dx 


y  =  i 


pa 
f    2pzdx 

J  V2px^dx 


=  a/P:^ 


X"'^ 


/  x^dx 


=  |4. 


2.  Find  the  centre  of  gravity  of  the  area  of  an  elliptic 
quadrant  whose  equation  is 

y  =  ^  Va»  —  a^. 


Here 


.      -       4« 


(1) 


rea  CAfiD. 


)f  a  semi-parab- 
linate,  then  we 

xdx 

—  =1*. 
*dx 

%  of  an  elliptio 


^xdx 
)^dx  ' 


MXAMPLtCa. 
4» 


117 


.-.    y  = 


3^' 


Hence  for  the  centre  of  gravity  of  the  area  of  a  circnlar 
quadrant  we  have 

-       -       4<i 

3..  Find  the  centre  of  gravity  of  the  area  of  a  semi- 
cycloid. 

Take  the  axis  of  the  cnrve  as  axis  of  ar,  and  a  tangent  at 
the  highest  point  as  axis  of  y ;  then  the  equation  is  (AnaL 
Geom.,  Art  167), 

y  =  0  vers-*-  +  's/'Hax  —  a*] 

where  a  is  the  radius  of  the  generating  circle.   Prom  (1)  wo 
have 


CD  = 


J^ydx        [yx-fxdy'^ 

\yx -  /(2ax  -  3?)*dx'^  *  TO.  2a  -  4»ra» ' 

since  when  x  =  0  and  2a,  y  =  Oanina. 

Also, 


■i  , 


Z' 


118  POLAB  ELSMElfTS  01^  A  PLANE  AJIBA. 


[y»a:-3  J^y  (^csr  -  «»)*  dxT 


37ra* 


[fx-^J*{2ax~a?')i  yevr^-ch—%f{^tax-7?)  dx'f 


37r«« 


[fx-%a3?4-\:t»  —  %nj{%ax  —  a*)i  verg-'  -  dxT 


arrfa» 


3  2 


3»ra» 


3ira» 


a 


which  the  student  can  verify  by  asBuming 

vers"*  -  =  fl. 
a 

(See  Todhunter's  Statics,  p.  118.) 

80.  Polar  Btomenta   of  a   Piano 

Area. — Let  AB  be  the  arc  of  a  curve, 
and  let  it  be  required  to  find  the  centre  of 
gravity  of  the  area  bounded  by  the  arc 
AB  and  the  extreme  radii-vecton*,  OA 
and  OB,  drawn  from  the  pole,  0,  to  the 
extremities  of  the  arc. 

Divide  the  area  into  infinitesimal  trianj^les,  such  as  POQ, 
included  between  two  oontrtLfCutive  radii-vectors,  OP  and 
0(j.  I^t  {r-9)  be  the  point,  P,  then  the  area  of  the 
element,  POQ  =:ft^c^  (Cal.,  Art.  191) ;  and  if  the  thick- 
ness and  density  of  the  lamina  are  uniform,  the  centre  of 


K19.41 


UtX—7?)  d£ 


a     Jo 


r 


such  as  POQ, 
itoni,  OP  and 
e  area  of  the 
if  the  thiok- 
the  centre  of 


MXAMPtm. 


in 


gravity  of  this  elementary  triangle  will  be  on  a  straight  lino 
drawn  from  0  to  the  middle  of  PQ,  and  at  a  distance  of 
two-thirds  of  this  straight  line  from  0  (Art  73).  Hence 
the  co-ordinates  of  the  centre  of  gravity,  g,  of  POQ,  are 
OM  and  Mi/,  or, 

fr  cos  0,    and    \r  sin  6. 


Hence,  (Art  77), 

-       /|rcosgjJfJ<?9  /f*oo80<l9. 

,       /frsing-jH^W  „  ,/r'singtJ9. 
"""  J-\f*dO         ""*     J'r*dd      ' 

the  integrations  extending  over  the  whole  area,  AOB. 


(1) 
(2) 


EXAMPLB. 

Find  the  centre  of  gravity  of  the  area  of  a  loop  of  Ber- 
nonilli's  Lemniscate  whose  equation  is  r*  =  «^  cos  5W. 

As  the  axis  of  the  loop  is  symmetrical  with  respect  to 
the  axis  of  a?,  y  =  0,  and  the  abscissa  of  the  centre  of 
gravity  of  the  whole  loop  is  evidently  the  same  as  that  of 
the  half-loop  above  the  axis.  Substituting  in  (1)  for  r  its 
value  a  cos^  20,  we  have 


•  =  fa 


X"' 


cost  20  cos  9  (i9 


X' 


cos  20  do 


s=  \aj{)  —  2  sin*  0)<  d  siii  0. 


Put  sin  0  = -7?,  then 


lao 


DOUBLB  IXTKOBATTOy. 


_  Ja     n^^^^  ^  J^  .  I J  (CaL,  Art.  167). 


3^/2 


•  = 


Tta 


4V^ 


81.  Double  Integrattoa— Polar  Fonniil«.— When 
the  density  of  the  lamina  varies  from  point  to  point,  it  may 
be  BeoesBary  to  divido  it  into  el. met  '  the  isecond  order 
instead  of  rectangular  or  triar^tu;.>  aements  of  the  first 
order  (Arts.  79  and  80). 

Suppose  that  the  density  of  the  hunina  AOB  (Fig.  41), 
is  not  nniform.  If  we  divide  it  into  triangular  eletnents, 
POQ,  the  element  of  mass  will  be  no  longer  proportional  to 
the  element  of  area,  POQ  =  ^i*d9',  nor  will  the  centre  of 
gravity  of  the  triangle,  POC*  ^  ^  distant  from  0. 

Let  a  series  of  circles  be  described  with  0  as  a  centre, 
the  distance  between  any  two  successive  circles  being  dr. 
These  circles  will  divide  the  triangle,  POQ,  into  an  infinite 
number  of  rectangular  elements,  abed  =  rdBdr.  If  k  is 
the  thickness  and  p  is  the  density  of  the  lamina  at  this  ele- 
ment, the  elemeii'  of  mass  will  hedtn  —  kprdSdr;  and 
the  coH>rdinates  of  its  centre  of  gravity  will  b  ''  ri?9  6  and 
r  sin  0,    Hence,  from  (1)  and  (3)  of  Art  '/7.       f  n'* 


m  = 


and 


r  fk  pr  done  rdBdr        f  fkpr*  cop  &  n'J  -  r 
ffhr  de  dr  ff^  ^  *''* 


V  — 


/A'* 


Bin  e  de  dr 


//^ 


d»dr 


(1) 


(•) 


In  each  of  these  integrals  the  values  of  k  and  p  are  to  be 
snbstitnted  in  terms  of  r  and  6,  and  the  intQgrstiona  taken 
between  proper  limits. 


3al.,Art.l57). 


nnlaB.— When 

I  point,  it  may 

Hocond  order 

ts  of  the  first 

OB  (Fig.  41), 

ular  eletaientfl, 

>roportional  to 

the  centre  of 

>mO. 

as  a  centre, 
^les  being  dr. 
tto  an  infinite 
ddr.  If  k  is 
oa  at  this  ele- 
prdBdr;  and 
> 


K 


'  fi?»  8  and 


6  rrj  r.r 

-     ,      (1) 
)dr 


(2) 


d  p  are  to  be 
rations  taken 


MXAMPLM. 


EXAMPLB. 


m 


Find  the  centre  of  gravity  of  the  area  of  a  cardioid  in 
which  the  density  at  a  point  increases  directly  as  its  distance 
from  the  cusp. 

Let  ft  =  the  density  at  the  unit's  distance  from  the 
cusp,  then  p  =  (ir,  is  the  density  at  the  distance  r  from 
the  cusp. 

As  the  axis  of  the  curve  h  an  axis  of  symmetry  (Art  67), 
y  =  0,  and  the  abscissa  ol  the  whole  curve  is  the  same  as 
for  the  halfabove  the  axis ;  then  (1)  becomes 


w  = 


=  } 


f     /f*cmedddr 
r  rt*dedr 

Jr*0Med6 


fr*dB 


by  performing  the  reintegration. 
The  equation  of  the  curve  is 


d 


r  s  a  (1  +  COB  9)  =  2a  oo*»  5 


Substituting  this  value  for  r,  and  putting  ^  =  0,  we  have 


2  s:  la 


J   00^  ^  (2  cos*  ^  —  1)  «^ 


/   oo^^d^ 


=  H«. 


in 


RXCTJLNaVLAB    FORMULAE. 


82.  DonUe  Integntioii.— Reotugiilar  Fommte.— 

Let  »  aeries  of  consecative  straight  lines  be  drawn  parallel 
to  the  axea  of  x  and  y  respectively,  diTidiog  the  area,  ABOD, 
(Fig.  40),  into  an  infinite  nnmber  o<'  rectangular  elements 
of  the  second  order.  Then  the  area  of  each  element,  as 
abed,  —  d^dy;  and  if  k  and  p  are  the  thickness  and  density 
of  the  lamina  at  this  element,  the  element  of  man  will  be 
dm  s=  kpdxdy,  and  the  co-ordinates  of  its  centre  of  gravity 
will  be  X  and  y.  Uenoe  from  (1)  and  (2)  of  Art  77,  we 
have 


•  = 


9  — 


J  jk (Kcdxdif 
yykpdxdy 

JJhpydxdy 


J  I  lepdxdy 
the  integrations  being  taken  between  proper  limita 


(1) 


(») 


EXAMPLE 

Find  the  centre  of  gravity  of  tVa  area  of  a  cycloid  the 
density  of  which  varies  as  the  nth  power  of  the  distance 
from  the  base. 

Take  the  base  as  the  axis  of  x  and  the  starting  point  as 
the  ori^!ii.    Then  the  equation  of  the  curve  is 

•  s  a  vers-*^  -  (Say  -  y»)* ; 


dx- 


V2ay- 


r  Fonnnlie.— 

drawn  parallel 
le  area,  ABOD, 
^lar  elements 
ich  element,  as 
ess  and  density 
f  man  will  be 
sntre  of  gravity 
of  Art  77,  we 


(1) 


(2) 


limits. 


a  cycloid  tbe 
the  diatance 

rting  point  as 


aUBFACa    OP  RSVOLUnON. 


123 


Let  p  =  ^y"  =  density  at  the  distanoe  y  from  the  base. 
It  is  evident  that  tbe  centre  of  gravity  will  be  in  the  axis  of 
the  cycloid  ;  therefore  i  =  rra  ;  and  as  it  is  constant  (3) 
becomes 


y  = 


J^J^trdydx 
n  +  It/p * 


_n  +  l^o     V2ay  —  y« 


«-»-2*  n  +  3  " 


'v/2ffy  —  y» 

rtf***  dy 


V2ffy  — y«. 


r 


0     V2«y  —  p 


y  = 


n_-f_l    2n  +  5 
n  +  2"  n  +  3  ' 


83.  Centre  of  OniTitjr  of  •  Bnrfiuse  of  Revola- 
tion. — Let  a  surface  be  generated  by  tbe  revolution  of  the 
curve,  aB  (Fig.  40),  round  the  axis  of  x.  Then  the 
elementary  arc,  PQ,  (=  ds),  generates  an  element  of  the 
nirf«o«  whose  aree  =  Stry  di  (Gal.,  Art  1.93).  It  k  ia  the 
tbicknoB8  and  p  the  density  of  the  lamina  or  shell  in  this 
elementary  zone,  the  element  of  mass  will  be  dm  =  %nkpy  ds. 
Also  the  centre  of  gravity  of  this  zone  is  in  the  axis  ul  x  at 


134 


SXAMPLSa. 


the  point  M  whose  absoiaaa  is  x  and  ordinate  0.    Hen<»  (1) 
of  Art.  77  becomes,  after  cancelling  *in, 


(1) 


Jkpxy  da 

X  :=  — — __ 

Jkpydt 
(.he  integrations  being  taken  between  proper  limits. 

BXAMPLSS. 

1.  Find  the  centre  of  gravity  of  the  surface  formed  by 
the  revolution  of  a  semi-cycloid  round  its  base. 
The  equation  of  the  generating  curve  is 


a 


»  =z  a  vers-*  *  —  \^^y  —  y*; 

d» 


or 


V3a-y 


which  in  (1)  gives,  after  cancelling  's/'ia  kp. 


2.  Find  the  centre  of  gravity  of  the  sarteoe  formed  by 
the  revolution  of  a  semi- cycloid  round  its  axis. 

It  is  clear  that  the  centre  of  gravity  lies  on  the  axis  of 
the  curve ;  hence  y  =  0.  ' 


0.    Hence  (1) 


(1) 


[imits. 


ace  formed  b; 
e. 


e  formed  by 
D  the  axis  of 


MXAMPLM8. 

The  equation  of  the  generating  carve  is 


135 


y  =  a  yerr->  -  +  V-Sox  —  a». 
a 


Here 


which  in  (1)  gires 


ds  =  VZax'^dx, 
J^ifxidx 


m  = 


J^y7r\ix 

[lya;*  —  \fx's/%a  —  x  dxT 
\%yx^  —  2 /V2a  — xifoT* 

_|ff<i(2a)*-- AW* 
""    »Tr  (2o)*  —  |(3a)» 


=  A« 


15tr-8 


8.  Find  the  centre  of  grayity  of  the  sarfaoe  formed  by 
the  revelation  of  the  semi-cycloid  round  the  axis  of  y  in  the 
last  example,  t.  e.,  round  the  tangent  to  the  carve  at  the 
highest  point 


Ah8.  P  =  jg  (ISrr 


8). 


i  iSii 


tse 


ANT   CUBVBD   BVaFACK. 


M.  Centre  of  Ghra-vity  of  Any  Onnred  Sarface.— 

Ixjt  there  be  a  shell  having  any  given  curved  surface  for 
one  of  its  boandaries;  and  let  *  =  tb?  thickness,  p  =  the 
density,  and  ds  —  the  area  of  an  element  of  the  surface  at 
the  point  {x,  y,  z);  then  (1)  of  Art.  83  becomes 


/  kpxda 


(1) 


and  similar  expressions  for  y  and  i. 

Substituting  the  value  of  ds  (CaL,  Art  201)  and  cancel- 
ling k  and  p,  we  have 

r  r  I,    ,  difi   ,   rfa»\i 


X  = 


ds* 


ffh%-'4f-^y 


BX  AM  PLE8. 


1.   Find    the  centre  of   gravity  of  one-eighth  of    the 
surface  of  a  sphere. 


Here 


«» +  y*  +  «^  =  a«. 


V    "^  «te»  "^  dyV    -  (o»  -  «»  -  y»)** 

rr      ^dxdy 
-  —  ^  ^  (<i»  -  ;>^  -  fy 

**'•"/*  r <^dy * 

J  J  {a*-^  r*  -  y«)* 


I  BnifAoe.— 
!<i  surface  for 
ness,  p  =  the 
he  surface  at 


(1) 


I  and  caucel- 


\> 


;hth  of   the 


30i«/D  or  aUVOLOTION. 


137 


First  perform  the  y-inljegn- 
tion,  X  being  constant,  firom 
y  ^0  to  y  =  -W  =  Sfi  = 
\/«*  —  a^ ;  the  effect  will  be 
to  sum  up  all  the  elements 
similar  to  pq  from  H  to  I. 
The  eflbct  of  »  subsequent 
a;-integration  will  be  to  sum 
all  these  elemental  strips  that 
are  comprised  in  the  surface 
of  which  OAB  is  the  projec- 
tion, and  the  limits  of  this  iutegratiou  are  x  =  0  and 
X  =  OA  =z  a.    Hence  t 


Fig.  ♦?. 


a  = 


Jo  Jn    (a' 


xdxdy 


»»)* 


P»  /f' dxdy 

Jn  Jq    (a»  -  a»  -  y»)* 


J^rxdz 


id. 


Similarly 


P  =  fi,    i  =r  i<i. 


2.  Find  the  centre  of  gravity  of  onn-eighth  of  the  sorfiMM 
of  the  sphere  if  the  density  varies  as  the  «-ordinate  to  any 
point  of  it.    Here  p  =  fix. 

.       .       4a     .       4a     .      3a 

J,«.  .  =  3-;  ,  =  _;.  =  -. 

85.  Centre  of  Qtmvity  of  *  Solid  of  Revolution. — 

Let  a  solid  be  generated  by  the  revolution  of  the  curve,  AB» 
(Fig.  40),  round  the  axis  of  x.  Then  the  clemeutury 
rectangle,  PQNM,  {=yflx),  gen^raitea  m  element  q£  th» 


198 


SOLID  or  RBVOLVnON. 


solid  whose  volnmc  =  n^dx  (Oal,  Art.  908).  Henoe  if  the 
density  of  the  solid  is  aniform,  we  have  for  the  position  of 
the  centre  of  gravity  ^whlch  evidently  is  in  the  axis  of  x), 


I  iTi/hi  dx        I  ^xdx 


^dx 


>dz 


(1) 


the   int^rations   being  extended   over   the  whole  area, 
CABD,  of  the  bounding  curve. 

If  the  density  varies,  the  element  of  mass  may  require  to 
be  taken  differently.  If  the  density  varies  with  z  alone,  t.  c, 
if  it  is  uniform  all  over  the  rectangular  strip,  PQNM,  the 
volume  may  be  divided  up  as  already  done,  and  the  element 
of  mass  =  Ttfyj^  dx.    Hence,  we  shall  have  v\  this  case, 


m  = 


Jpj/*xdx 

— — — ■—     ■  '  ■-— -  • 

fpfdx 


(2) 


If  the  density  varies  as  y  alone,  we  may  take  a  rectangular 
element  of  area  of  the  second  order,  dx  dy,  at  the  point 
(x,  y) ;  this  area  will  generate  an  element  of  volume 
=  2ny  dx  dy ;  therefore  the  element  of  mass  =  2irpy  dx  dy, 
and  we  have 


»  = 


JJpxydx  :y 
fj'pydxdy 


(8) 


the  y-integrations  being  performed  first,  from  0  to  y,  the 
ordinate  of  a  point  P,  on  the  bounding  curve ;  and  then 
the  x-integrations  from  OC  to  OD. 


mm' 


8).  Henoe  if  the 
»r  the  position  of 
I  the  axis  of  x), 


(1) 


the  whole  area, 

18  may  require  to 
with  «  alone,  t.  0., 
rip,  PQIf^M,  the 
and  the  element 
in  this  case, 


(«) 


ke  a  rectangular 
iff,  at  the  point 
lent  of  volame 
=  2iTpy  dx  dy, 


(8) 


■om  0  to  y,  the 
iirve ;  and  then 


BXAMPLSa. 


SXAMPLE8. 


189 


-  1.  Find  the  centre  of  gravity  of  the  hemisphere  generated 
by  the  revolution  of  the  quadrant,  AD,  (Fig.  39),  round  OA 
(taken  as  axis  of  x),  (1)  when  the  density  is  uniform ;  (2) 
when  it  is  constant  over  a  section  perpendicular  to  OA  and 
varies  as  the  distance  of  this  section  from  OD ;  (3)  when 
it  is  constant  at  the  same  distance  from  CA  And  varies  as 
this  distance. 


(1)  From  (1)  we  hare 


X  = 


jj/hidx 
J  y*dx 


Putting  35  =  r  cos  6,  and  y  =  r  sin  e,  where  r  is  the 
i-adius  of  the  circle   and  integrating  hetween  0  =  0  and 


e 


2' 


we  have 


(2)  Since  p  =  fia;,  we  have  from  (2) 

faN/»dx 


w  = 


which  gives 
(3)  Since  p 


»  =: 


~m  -  tV- 
Hy,  we  have  from  (3) 

r  Ixj^dxdy         Cxf^dx 
ffy*dxdy        fi^dx' 


180 


axAMPLm. 


■•,  If 


aatl  the  previons  Bnbstituv^ons  for  x  and  y  give 

16r 


u  = 


ISir 


2.  Find  the  centre  of  gravity  of  a  paraboloid  of  revolu- 
tion, the  length  of  whose  uxis  is  A.  Ant.  i  =  \h. 

3.  Find  the  centre  of  gravity  (1)  of  a  portion  of  a  prolate 
spheroid,  the  length  of  whose  axis  measured  from  the  vertex 
is  c,  and  (2)  of  a  heini-spheroid. 

An».  (1)  i  =  I  -g— ~;  (2)  *  =  |a. 

66.  Polar  Fonnnlie.—  a  solid  be  generated  by  the 
revelation  of  AB,  (Fig.  41,  1  the  axis  of  x.    Then  the 

elementary  rectangle,  ahcd,  whose  mass  =  pr  dd  dr,  (Art 
81),  the  thickness  being  omitted,  generni'ts  a  ring  which  is 
an  element  of  the  solid  whose  volume  =  %nr  sin  0  pr  dd  dr ; 
and  the  abscissa  of  the  centre  of  gravity  of  the  ring  is 
r  cos  0.    Hence  (1)  of  Art.  77  becomes 


r  jp^  sin  0  cos  0  (/d  dr 

~7P- 


sin  eMdr 


(1) 


in  which  p  mnst  be  a  function  of  r  and  6  in  order  that  the 
integrations  may  be  effected. 

If  the  density  depends  only  on  the  distance  from  a  fixed 
point  in  the  axis  of  revolution,  this  point  may  be  taken  as 
origin,  and  p  will  be  a  function  of  r ;  if  the  density  depends 
only  on  the  distance  from  the  axis  of  revolution,  p  will 
be  a  function  of  r  sin  6. 


IXAMPLK. 


The  vertex  of  a  right  circular  cone  is  in  the  sur&ce  of  a 
sphere,  the  axis  of  the  cone  coinciding  with  a  diameter  :>f 


ygire 


rsboloid  of  revolu- 
Ang.  i  =  |A. 

wrtion  of  a  prolate 
rod  from  the  vertex 

I  generated  bj  the 
18  of  X.  Then  the 
=  pr  dO  dr,  (Art 
5^8  a  ring  which  is 
Inr  sin  6  pr  dS  dr ; 
ty  of  the  ring  is 


dr 


(1) 


in  order  that  the 

nee  from  a  fixed 

may  be  taken  as 

density  depends 

evolution,  p  will 


the  surface  of  a 
th  a  diameter  c»f 


OENTRK  OF  ORAVtTT  Of  AITT  80UO. 


131 


the  sphere,  the  base  of  the  cone  being  a  portion  of  the  sur- 
face of  the  sphere.  Find  the  distance  of  the  centre  of 
gravity  of  the  cone  from  its  vertex,  2a  being  its  vertical 
angle,  and  a,  the  radius  of  the  sphere. 

Here  the  r-limits  are  0  and  2a  cos  0 ;  the  0-limits  are  0 
and  « ;  p  is  constant ;  hence  from  (1)  wo  have 


a  = 


*'0  «^0        y 


sin  6  cos  0  dS  dr 


r  Ht*  sin  e  dd  dr 


=  1 


r(2fl  (,08  (?) 


*  ein  0  cos  0  d$ 


=  |« 


r(2a  cos  6)*  sin  0  d9 
/"co8^  e  sin  9  d9 


/■ 


cos*  0  sin  d  do 


1  —  coal*  a 
1  —  cos* 


a. 


87.  Centre  of  Gkavity  of  any  Solid.— Let  (x,  y,  z) 

and  {x  •{■  dx,  y  ■{■  dy,  z  +  dz)  be  two  consecutive  points  E 
und  F,  (Fig.  42),  within  the  solid  whose  centre  of  gravity  is 
to  be  found.  Through  E,  pass  three  planes  parallel  t<o  the 
co-ordinate  planes  xy,  yz,  zx ;  also  through  F  pass  three 
planes  parallel  to  the  first.  The  solid  included  by  these  six 
planes  is  an  infinitesimal  parallelopiped,  of  which  £  and  F 
are  two  opposite  angles,  and  the  volume  =  dx  dy  dz.  If  p 
is  the  density  of  the  body  at  £,  the  element  of  mass  at  E 
=  pdT  dy  dz.  Hence  the  co-ordinates  of  the  centre  of 
gravity  ot  the  solid  are  given  by  the  equations 


IM 


MXAMPLKS. 


m  = 


t  = 


J  J  J  pxdxdydt 
fffpdxdydz  ' 

J  jfpydxdydt 
/ffpdxdydz 

fffptdxdydB 


the  integrations  being  extended  over  the  whole  solid. 


(1) 


(^) 


(3) 


EXAMPLES. 


1.  Find  the  centre  of  gravity  of  the  eighth  part  of  an 
ellipsoid  included  between  its  three  principal  planes.* 

Let  the  equation  of  the  elliiMSoid  be 


^  +  |i  +  -^^i- 


Here  the  limits  of  the  ^-integration  are 
which  call  e,  and  0 ;  the  limits  of  y  are 
which  call  y,  and  0 ;  the  z-limits  are  a  and  0. 


*  FlMMt  of  xf .  y*.  w. 


1,  jjjgiwWJWKTww 


mm 


POLAR  KLKMSlfTS  OF  "    .J. 


188 


(1) 


(2) 


(3) 


irhole  solid. 


ighth  part  of  an 
pal  planes.* 


0. 


First  integrate  with  respect  to  z,  and  we  obtain  the 
infinitesimal  prismatic  colnmu  whose  base  is  PQ.  (Fig.  42), 
and  whose  height  is  Pp.  Then  we  integrate  with  respect 
to  y,  and  obtain  the  sum  oi*  all  the  columns  which  form 
the  elemental  slice  Kplinq.  Then  integrating  with  respect 
to  as,  we  obtain  the  snm  of  all  the  slices  included  in  the 
solid,  OABO.  Hence  (1)  becomes,  since  the  density  is 
uniform. 


rrr^'^^^y^ 


m  := 


/"y.'X'*"'** 


^^^•-^^*''^* 


n-^),>. 


.'.   »=s^a. 


Similarly         y  =  fft,    ■  =  |«. 

2.  Find  the  centre  of  gravity  of  the  solid  bonnded  by  th ' 
planes  »  =r  (ix,  n  =  yx,  and  the  cylinder  y»  =  2a«  —  i*. 

88.  Polar  El«m«iit»  of  MMm.— Lei  Fig.  43  repre- 
sent the  portion  of  the  volume  of  a  solid  included  between 
its  bounding  surfaov;  and  three  rectangular  oo-ordinato 
planon, 


'Ul 


134 


POLAR  MLEMSSTB  OF  MASS. 


mi 


(1)  Throngh  the  axis  of  z  draw 
a  aeries  of  consecutive  plaues,  divid- 
ing the  soUd  into  wedgenshc^d 
slices  such  as  OOBA. 

(2)  Round  the  axis  of  t  describe 
a  series  of  right  cones  with  their 
vertices  at  O,  thus  dividing  each 
slice  into  elementary  pyramids  like 
0-PQST. 

(3)  With  0  as  a  centre  describe 
a  series  of    consecutive  spheres; 
thus  the  solid  is  divided  into  elementary  rectangular  par- 
allelopipeds  similar  to  abpt,  whose  volume  —  ap-ps'  at. 

Let  XOA  =  <l>,    COP  =  e.    Op  =2  r, 

AOB  =  dit,  rOQ  =:de,  pa-  dr. 

Then  pq  is  the  arc  of  fe  circle  whose  radius  is  r,  and  the 
angle  is  de ;  therefore 

pq  =  rd9. 

Also  pa  is  the  furc  of  a  circle  in  which  the  angle  is  d<p, 
and  the  radius  is  the  perpendicular  from  p  on  OZ,  or 
r  sin  0;  tharefore 

ps  —  r  sin  e  d^. 

Therefore  the  volume  of  the  elementary  parallelopiped  = 

>^  sin  6  dr  dd  dp ; 

and  if  p  is  the  density  of  the  solid  at  p,  the  element  of 
mass  is 

pr*  »iu  e  dr  de  dp. 

Also  the  oo>ordiDates  of  the  centre  of  gravity  of   this 
element  are 

r  sio  0  ooe  ^,    r  ein  9  aiu  <P,    and    r  cos  9 ; 


ii'SM'f  f 


y  rectangttlar  par- 
'  ~  ap'ps'  at. 

=  r, 

-dr. 

ins  is  r,  and  the 


the  angle  is  d<l>, 
'm  p  on  OZ,  or 


>araIlelopiped  =r 
the  element  of 

rwvity  of   thii 
rco8ff; 


mmm 


HXAMPLSa.  135 

hence  for  the  centre  of  gravity  of  the  whole  solid  we  have 

/  /  fpf*  sin*  Ocoifclrddd^ 
J'fjp**  Bin  edrdddit 


a  = 


fffpr*  »in*  e  iia  ift  dr  de  d^ 
Jjffpf*  sin  edrded^ 

r  r  ipr*  sin  0  COB  0  dr  dO  d^ 


ti  = 


fffpf*  sin  Bdrded^ 


the  limit!  of  integration  being  determined  by  the  figure  of 
the  solid  considered. 

The  angles,  9  and  ^,  are  sometimes  called  the  eo4at%tud»t 
and  longitude,  respectively. 

BXAMPLB8. 

1.  Find  the  centre  of  gravity  of  a  hemisphere  whose 
density  varies  as  the  nth  power  of  the  distance  from  the 
centre. 

Take  the  axis  of  f  perpendicnlar  to  the  plane  bMC  of  the 
hemisphere.  Let  a  =  the  radios  of  the  sphere,  and 
p  =  fir*,  where  n  is  the  density  at  the  nnits  distance  from 
the  centre.  First  integrate  with  respect  to  r  from  0  to  a, 
and  we  obtain  the  infinitesimal  pyramid  0-PQST.  Then 
integrate  with  respect  to  0  from  0  to  ^n,  and  we  obtain  the 
sum  of  all  the  pyramids  which  form  the  elemental  slice, 
CODA.  Then  integrating  with  respect  to  ^  from  0  to  arr, 
we  obtain  the  sum  of  all  the  slices  included  in  the  hemi- 
sphere.    Heuce, 


/      /     Bin  (9  COS  0  rfd  d^ 
y      /    sin  0  rf0  0?^ 


i  =  P  =  0. 

2.  Find  the  centre  of  gravity  of  a  portion  <  f  a  solid 
sphere  conlaired  in  a  right  cono  whose  vertex  is  the  centre 
of  the  sphere,  the  density  of  the  solid  varying  as  the  «th 
power  of  the  distance  from  the  centre,  the  vertical  angle  of 
the  cone  being  =  2«,  and  the  radius  =  a. 

Take  the  axia  of  the  cone  as  that  of  t,  and  any  plane  through  it  as 
that  from  which  longitude  is  nieaaured. 

^*^-  •  =  ^      ^^(1  +  cos  e),  and  i  =  y  =  0. 

89.  8p«oi«l  Metfiods.— In  the  preceding  Articles  we 
have  given  the  usual  forraulw  for  finding  the  centres  of 
gravity  of  bodies,  bnt  particular  cases  may  occur  which  may 
be  most  conveniently  treat  U  by  special  methods. 

BXAMPLBS. 

1.  A  circle  revolves  round  a  tangent  line  through  an 
angle  of  180°  find  the  centre  of  gravity  of  the  solid 
generated. 


drd$dift 

iedi» 


i6d<tt 


r^ 


rtion  ( f  a  solid 
tex  is  the  centre 
ying  as  the  «th 
vertical  angle  of 

lane  through  It  aa 
I  *  =  y  =  0. 


ng  Articles  we 
the  centres  of 
icur  which  may 
uds. 


e  through  an 
of  the  solid 


BXAXPLKB, 


137 


Let  07  be  the  tangent  line  about 
•  which  tb<)  circle  revolves,  and  let  the 
plane  of  tbe  paper  bisect  the  solid ;  the 
centre  of  gravity  will  tberefora  lie  in 
the  axis  of  x.  Lot  P  and  Q  bo  two 
consecutive  points  :  and  let  OM  =  x, 
and  MP  =  y  =  V2«.c  —  »«.  The 
elementary  i-ectangle,  PQy;?,  will  gen- 
crate  a  semi-cylindrical  shell,  whose  volume  =  2ffnxdx, 
the  centre  of  gravity  of  which  will  be  in  the  axis  of  re  at  a 

distanoe  —  ftom  0  (Art  78,  Ex.  1,  Oor.).     Hence, 


r~-'> 


TTxdx 


JT'y 


nx  dx 


„  /     3^  V2ax  —  afidx 

A  I'D 

ir    /**«       ~ 

/     X  V  a«w  —  s^dx 


6a 

2^* 


3.  Find  the  centre  of  gravity  of  a  right  pyramid  of  uni- 
form density,  whose  base  is  any  regular  plane  figure. 

Let  the  vertex  of  the  pyramid  be  the  origin,  and  the  axis 
of  the  pyramid  the  axis  of  x;  divide  the  pyramid  into  slices 
of  the  thickness  dx  by  planes  perpendicular  to  the  axis. 
Then  as  the  areas  of  Ihnse  sections  are  as  the  squares  of 
their  homologous  sides,  and  as  the  sides  are  as  their  dis- 
tances from  the  vortex,  so  wiU  the  areas  of  the  sections  be  as 
the  squares  of  their  distances  from  the  vertex,  and  therefore 
the  masses  of  the  slices  arc  as  ^ho  squares  of  their  distances' 
from  the  vertex.  Now  imi^ne  each  slice  to  be  condensed 
into  its  centre  of  gravity,  which  point  is  on  the  a^ic  of  x. 
Then  the  problem  is  reduced  to  finding  the  centre  of  grav> 


1S8 


TBXORKMS  OF  PAPPUS. 


Jty  of  a  material  line  io  which  the  density  varies  as  the 
square  of  the  distance  from  one  end,  and  which  may  be 
found  as  in  Ex.  6,  (Art  78).  Calling  a  the  altitude  of  the 
pyramid,  we  have 


a^dz 


X  ■■= 


_  fa, 


which  if  the  same  as  in  Art.  75. 


■I 


I 


90.  TlMormiui  of  Pappus.*— (1)  If  a  plans  curve 
revolve  rourd  any  axis  in  its  plane,  the  area  of  the 
surface  generated  is  equal  to  the  length  of  the 
revolving  curve  multiplied  by  the  length  of  the 
path  described  by  its  centre  of  gravity. 

Let «  denote  the  length  of  the  carve,  x,  y,  the  co-ordinates 
of  one  of  its  points,  i,  y,  the  co-ordinates  of  the  centre  of 
gravity  of  the  curve;  then,  if  the  curve  is  of  constant 
thickness  and  decity,  we  have  from  (3)  of  Art.  78, 


y-  —71—; 
J" 

in  pa  =  Zn  /  yds; 


(1) 


the  second  member  of  which  is  the  area  of  the  surface 
generated  by  the  revolution  of  the  curve  whose  length  is  8 
about  the  axis  of  x,  (GaL,  Art.  193) ;  and  thf  first  member 
is  the  length  of  the  rovolviug  curve,  s,  multiplied  by  the 
length  of  the  path  described  by  its  centre  of  gravity,  2ffy. 

•  tJnkllr  oafled  Onldin'i  Tbeormm,  bnt  origlDally  •■randated  Ity  n^vpns.   0m 
Watton**  MeduBloai  Protdnaa,  p.  48,  Sd  W.) 


lity  varios  aa  the 
d  which  may  be 
le  altitade  of  the 


a  plane  eurve 
the  area  of  the 
length  of  the 
length  of  the 
'ty. 

the  co-ordinatea 
of  the  centre  of 
e  is  of  constant 

Art.  78, 


(1) 

of  the  surface 
lose  length  is  a 
u  first  member 
iltiplied  by  the 

gravity,  2»ry. 

Med  Iqr  Fipptu.   ifim 


wmm. 


wm 


wammm 


TBSORBMB  OF  PAPPUS. 


139 


(2)  //  a  plane  area  revolve  round  any  axis  in  its 
plane,  the  volume  generated  is  equal  to  the  area  of 
the  revolving  figure  multiplied  by  the  length  of  the 
path  described  by  its  centre  of  gravity. 

Let  A  denote  the  plane  area,  and  let  it  be  of  constant 
thickness  and  density,  then  {%)  of  Art  82  becomea 


V  = 


jfy  dx  djf 
ffdxd/ 

or  2?ry  y    fdA  ■=  %ir  J  jy  dx  dy, 

(substituting  dK  for  dx  dy), 


(») 


the  integral  being  taken  for  every  point  in  the  perimeter  of 
the  area;  but  the  second  member  is  the  volume  of  the 
solid  generated  by  the  revolution  of  the  area  (Oal.,  Art. 
208) ;  and  the  first  member  is  the  area  of  the  revolving 
figure,  A,  multiplied  by  the  length  of  the  path  described 
by  its  centre  of  gravity,  2ny. 

Cob. — If  the  carve  or  anaa  revolve  throngh  any  angle,  0, 
instead  of  2it,  (1)  and  (2)  become 


and 


6i8  =  efyd$, 
9yK  =  ^fi/»dx. 


itnd  the  theorem*  are  still  true. 


SoH. — If  the  axis  cuts  the  revolving  curve  or  area,  the 
theorems  still  apply  with  the  convention  that  the  surface 
or  volume  generated  by  the  portioni  of  the  curve  or  area  on 
opposite  sides  of  the  axis  are  affected  with  opposite  rignc. 


i*fl3! 


m 


140 


JSXAMPLMS. 


EXAMPLES. 

1.  A  circle  of  radios,  a,  revolves  round  ua  axis  in  its  own 
plane  at  a  distance,  c,  from  its  centre;  find  the  surface  of 
the  ring  generated  by  it. 

The  length  (circumference)  of  the  revolving  carve  = 
2na;  the  length  of  the  path  described  by  its  centre  of 
gravity  =  inc; 

.'.    the  area  of  the  surface  of  the  ring  =  in*ac. 

2.  An  ellipse  revolves  round  an  axis  in  its  own  plane, 
the  perpendicular  distance  of  which  from  the  centre  is  c  ; 
find  the  volume  of  the  ring  generated  daring  a  comph  t€ 
revolution. 

Let  a  and  b  be  the  semi-axes  of  the  ellipse ;  then  the 
revolving  area  =  nab ;  the  length  of  the  path  described  by 
itf(  centre  of  gravity  =  2nc ; 

.  • .    the  volume  of  the  ring  =  in'abc. 

Observe  that  the  volome  is  the  same  for  any  positioii  of  the  axes 
of  tlie  ellipse  with  respect  to  th«  axis  of  revolution,  provided  the  per- 
pendicular diiitance  from  that  axis  to  the  centre  of  the  ellipse  is  the 
same. 

3.  The  surface  of  a  sphere,  of  radius  a,  =  inefi;  the 
length  of  a  somi-circumference  =  na ;  find  the  length  of 
the  ordinate  to  the  centre  of  gravity  of  the  arc  of  a  semi- 
circle. ^       -       2a 


Ans. 


y   =  __. 


4.  The  volume  of  a  sphere,  of  radius  a,  =  fTro* ;  the 
area  of  a  semicircle  =  rra* ;  find  the  distance  of  the  centre 
of  gravity  of  the  semicircle  from  the  diameter. 


Aru.  y  = 


40 

37r* 


6.  A  circular  tower,  the  d''uneter  of  which  is  20  ft.,  is 
being  bnilt,  and  for  every  foot  it  rises  it  inclines  1  in.  from 


I  axis  in  its  own 
i  the  Bur&ce  of 

)lving  carve  = 
ly  its  centre  of 

its  own  plane, 
the  centre  is  c ; 
ing  a  complete 

lipse;  then  the 
th  described  by 

idtion  of  the  axes 
provided  the  per- 
the  elllpoe  ia  the 

,  =  4rro*;    the 

the  length  of 

arc  of  a  ecmi- 

-       2a 
ns.  y  =  — . 

It 

=  ^ira' ;  the 
of  the  centre 

4tf 

"*•  ^  =  rn 

h  is  20  ft.,  is 
nes  1  in.  from 


JSXAMPLSa. 


141 


the  vei:tical ;  find  the  greatest  height  it  can  reach  without 
falling.  Ans.  240  ft. 

6.  A  circular  table  weighs  20  lbs.  and  rests  on  four  tegs 
in  its  circumference  forming  a  square  ;  find  the  least  ver- 
tical pressure  that  must  be  applied  at  its  edge  to  overturn  it. 

Ans.  20  {V2  +  1)  =  48.28  lbs. 

7.  If  the  sides  of  a  triangle  be  3,  4,  and  5  feet,  find  the 
distance  of  the  centre  of  gravity  from  each  side. 

Ans.  I,  1,  f  ft. 

8.  An  equilateral  triangle  stands  vertically  on  a  rough 
plane  ;  find  the  ratio  of  the  height  to  the  base  of  the  plane 
when  the  triangle  is  on  the  point  of  overturning. 

Ans.  V3  :  1. 

9.  A  heavy  bar  14  feet  long  is  bent  into  a  rigbc  angle  so 
that  the  lengths  of  the  portions  which  meet  at  t.ie  angle 
are  8  feet  and  6  feet  respectively  ;  show  that  the  distance 
of  the  centre  of  gravity  of  the  l»r  so  bent  fh)m  the  point 
of  the  bar  which  was  the  centre  of  dravity  when  the  bar 

9  V^ 
was  straight,  is  — ^ —  feet. 

10.  An  equilateral  triangle  rests  on  a  sqnare,  and  the  base 
of  the  triangle  is  equal  to  a  side  of  the  square  ;  find  the 
centre  of  gravity  of  the  fligure  thus  formed. 

Ans.  At  a  distance  from  the  base  of  thn  triaugle  equal  to 

7=  of  the  base. 

8  +  2^3 

11.  Find  the  inclination  of  a  rough  plane  on  which  half 
p.  i^gnkiT  hexagon  can  just  rest  in  a  vertical  position  with- 
out overturning,  with  the  shorter  of  its  parallel  sides  in 
contact  with  the  plane.  Ans.  3  y/Z  :  5. 

12.  A  cylinder,  the  diameter  of  which  is  10  ft,  and  height 
60  ft,  rests  on  another  cylinder  the  diameter  of  which  is 


■i*ii 


143 


EXAMPLMS, 


18  fL,  and  height  6  ft ;  and  their  axes  coincide ;  find  their 
common  centre  of  gravity.  Ans,  273|f  ft.  from  the  base. 

13.  Into  a  hollow  cylindrical  vessel  11  ins.  high,  and 
weighing  10  lbs.,  the  centre  of  gravity  of  which  is  5  ins. 
from  the  bai>e,  a  uaiform  sohd  cylinder  C  ins.  long  and 
weighing  20  lbs.,  is  just  fitted ;  find  their  common  centre  of 
gravity.  Ans.  3f  ins.  from  base. 

14.  The  middle  points  of  two  adjacent  sides  of  n  square 
are  joined  and  the  triangle  formed  by  this  straight  line  and 
the  edgep  is  cut  ofF;  find  the  centre  of  gravity  of  the 
remainder  of  the  square. 

Ans.  ^ot  diagonal  from  centre. 

15.  A  trapezoid,  whose  parallel  sides  are  4  and  12  ft. 
long,  and  the  other  sides  each  equal  to  5  ft.,  is  placed  with 
its  plane  vertical,  and  with  its  shortest  side  on  an  inclined 
plane ;  find  the  relation  between  the  height  and  base  of  the 
plane  when  the  trapezoid  is  on  the  point  of  falling  over. 

Ans.  8  :  7. 

16.  A  regular  hexagonal  prism  is  placed  on  an  inclined 
plane  w:th  its  end  faces  vertical ;  find  the  inclination  of 
the  plarie  so  that  the  prism  may  just  tumble  down  the  plane. 

Ans.  30". 

17.  A  regular  polygon  just  tumbles  down  an  inclined 
plane  whose  inclination  is  10° ;  how  many  sides  has  the 
polygon  ?  Ans.  18. 

18.  Prom  a  sphere  of  radius  R  is  removed  a  sphere  of 
radius  r,  the  distance  between  their  centres  being  c;  find 
the  centre  of  gravity  of  the  remainder. 

Ans.  It  is  on  the  line  joining  their  centres,  and  at  a  dis- 
cr* 


tance 


/2«-r» 


from  the  centre. 


19.  A  rod  of  uniform  thickness  is  made  np  of  equal 
lengths  of  three  sabstauoes,  the  densities  of  which  taken  in 


icide;  find  their 
from  the  base. 

1118.  high,  and 
which  is  5  iiis. 
6  ins.  long  and 
foimon  centre  of 
na.  from  base. 

des  of  a  square 

traight  line  and 

gravity  of  the 

from  centre. 

e  4  and  12  ft. 
,  is  placed  with 
on  an  inclined 
and  base  of  the 
falling  over. 
J««.  8  :  7. 

on  an  inclined 

i  inclination  of 

iown  the  plane. 

Ans.  30°. 

vn  an  inclined 
sides  has  the 
Ans.  18. 

'ed  a  sphere  uf 
i  being  e ;  find 

,  and  at  a  dis- 


•  tip  of  equal 
rbich  taken  in 


tXAMPLWa. 


143 


order  are  in  the  proportion  of  1,  %,  and  3  ;  find  the  position 
of  the  centre  of  gravity  of  the  rod. 

Ans.  At  -^  of  the  whole  length  from  the  end  of  the 
densest  piu*t. 

20.  A  heavy  triangle  is  to  bo  suspended  by  a  string  pass- 
ing through  a  point  on  one  side ;  determine  the  position  of 
the  point  so  that  the  triangle  may  rest  with  one  side 
vertical. 

Ans.  The  distance  of  the  point  from  one  end  of  the  side 
=r  twice  its  distance  from  the  other  end. 

21.  The  sides  of  a  heavy  triangle  are  3,  4,  5,  respectively ; 
if  it  be  suspended-  from  the  cetitre  of  the  inscribed  circle 
show  that  it  will  rest  with  the  shortest  side  horizontal. 

22.  The  altitude  of  a  right  cone  is  h,  and  a  diameter  of 
the  base  is  i ;  a  string  is  fastened  to  the  vertex  and  to  a 
point  on  the  circumference  of  the  circular  base,  and  is  then 
put  over  a  smooth  peg ;  show  that  if  the  cone  rests  with  its 
axis  horizontal  the  length  of  the  string  is  ^/{Ifl  +  ¥). 

23.  Find  the  centre  of  gravity  of  the  helix  whose  equa- 
tions are 

X  =■  a  cos  ^  ;    y  =  a  sin  0  ;    «  =  ha^. 


Ans.  X 


,    y    .        J    a—x    -       % 
ka^;  y  =  ka  -— -;  z  =  ^y 

Z  z  a 


24.  Find  the  distance  of  the  centre  of  gravity  of  the 
catenary  (Cal.,  Art  177),  from  the  axis  of  x,  the  curve 
being  divided  into  two  equal  portions  by  the  axis  of  y. 

Ans.  If  2Z  is  the  length  of  the  curve  and  (/*,  k)  ig  the 
extremity,  the  centre  of  gravity  is  on  the  axis  of  y  at  a 

distance  — ^ —  trom  the  axis  of  x. 


KXAMPLBS. 

25.  Find   the  centre  of  gravity  of  the  area  included 
between  the  arc  of  the  parabola,  y»  =  4aa:,  and  the  straight 


line  y  =  kx 


Arts,  i  = 


?1 


y  = 


2rt 


26.  Find  the  centre  of  gravity  of  the  area  bounded  by 
the  cissoid  and  its  asymptote,  the  equation  of  the  cissoid 
being  y»  =  --^--.  ^^.  ^  ^  j^, 

27.  Find  the  centre  of  gravity  of  the  area  of  the  witch 
of  Agnesi. 

Afis.  At  a  distance  from  the  asymptote  equal  to  ^  of  the 
diameter  of  the  base  circle. 

28.  Find  the  centre  of  gravity  of  the  area  included  be- 
tween the  arc  of  a  semi-cycloid,  the  circumference  of  the 
generating  circle,  and  the  base  of  the  cycloid,  the  common 
tangent  to  the  circle  and  cycloid  at  the  vertex  of  the  latter 
being  taken  as  axis  of  z,  the  vertex  bein^r  origin,  and  a  the 
radius  of  the  generating  circle. 

.        .37r»  —  8        _ 
Ans.  x=  —^--— a;  y  ==  in. 

39.  Find  the  centre  of  gravity  of  the  area  contained  be- 
tween the  curves  y'  =  ax  and  f  =  2ax  —  a^,  which  is 
above  the  axis  of  a;.    ,       .  15  n-  —  44 

y  = 


Atis.  i  =  o 


a 


IStt  —  40' 


3n  -  8 


30.  Find  the  centre  of  gravi>;^  of  the  area  included  by 
the  curves  y*  =  ax  and  a^  =  ly. 

Ana.  i:  =  ^ijl ;  y  —  ^ibK 

31.  Find  tno  distance  of  the  centre  of  gravity  of  the  area 
of  the  circular  sector,  BOCA,  (Fig.  39),  from  the  centre. 

Let  20  =  the  angle  included  by  the  bounding  radii 

.„„   -       .sin  © 
Ans.  «  =  f*  -^-. 


SXAMPLSa. 


146 


area  inclndod 
ind  tho  straight 
8a      _  _  2rt 

ea  bonnded  by 
I  of  tho  cissoid 

ins.  i  =  \a. 
a  of  the  witch 
[ual  to  I  of  the 

[.  included  be- 
ference  of  the 
,  tho  common 
of  the  latter 
^n,  and  a  the 

y  —  {a. 

contained  be- 
■  «*,  which  is 

—       ^ 
included  by 

=  A«***. 

_,  of  the  area 
:he  centre. 
g  radiL 
,    sin  9 


32.  Find  the  distance  of  the  centre  of  gravity  of  the 
circular  segment,  BCA,  (Fig.  39),  from  the  'centre. 

.        -       ,  a  sin'  B 

Ana.  «  =  f 


BC' 


»  —  sin  «•  cos  e  ~  12  urea  of  ABC 

33.  Find  the  centre  of  gravity  of  the  area  bounded  by 
the  cardioid  r  =  o  (1  +  cos  6).  Ana.  i  =  ^a. 

34.  Find  the  centre  of  gravity  of  the  area  included  by  a 
loop  of  the  curve  r  ==  a  cos  20.  _       28a  Vi 


Ans.  X  = 


105n- 


35.  Find  the  centre  of  gravity  of  the  area  included  by  a 
loop  of  the  curve  r  =  a  cos  3  .  _       gja  y^ 


Ana.  te  = 


80rr 


36.  Find   the  centre  of   gravity  of   the  area  of   the 
sector  in  Ex.  31,  if  the  density  varies  directly  as  the  dis- 


tance from  the  centre. 


Ana.  X  = 


3a    sin  6 


e 


37.  Find  the  centre  of  gravity  of  the  area  of  a  circular 
sector  in  which  the  density  varies  as  the  «th  power  of  the 
distance  from  the  centre. 

Ana.  — ^t_  .  _,  where  a  is  the  radius  of  the  circle,  I  the 
fi  +  o      I 

length  of  the  arc,  and  c  tue  length  of  the  chord,  of  the 

sector. 

88.  Find  the  centre  of  gravity  of  the  area  of  a  circle  in 
which  the  density  at  any  point  varies  as  the  nth  power-of 
the  distance  from  a  given  point  on  the  circumference. 

Ana.  It  is  on  tho  diameter  passing  through  the  given 

2(n  + 


point  at  a  distance  from  this  point  equal  to 
a  being  the  radius. 


«  +  4 


<h 


149 


SXAMPLSa. 


39.  Find  the  centre  of  gravity  of  the  area  of  a  quadrant, 
of  an  ellipse  in  which  the  density  at  any  point  yaries  as 
tiie  distance  of  the  point  from  the  major  axis. 


Ans.  ii  =  |a  ;  y 


3?r. 


40.  Find  the  distance  of  the  centro  of  gravity  of  the  sur- 
face of  a  cone  from  the  vertex. 


Let  a  =  the  altitude. 


Ans. 


K 


41.  Find  the  centre  of  gravity  of  the  surface  formed  by 
revolving  the  curve 

r  =  o  (1  -f-  cos  6), 


round  the  initial  I'ne. 


.       -       60a 
Ans.  •  =  -gg . 


42.  A  parabola  revolves  round  its  axis ;  find  the  centre 
of  gravity  of  a  portion  of  the  surface  between  the  vertex 
and  a  plane  perpendicular  to  the  axis  at  a  distance  from 
the  vertex  equal  to  J  of  the  latus  rectum. 

Ans.  Its  distance  from  the  vertex  =  f  |  (latns  rectum). 

43.  Find  the  centre  of  gravity  of  a  cone,  the  density  of 
each  circular  slice  of  which  varies  as  the  nth  power  of  its 
distance  from  a  parallel  plane  through  the  vertex. 

Let  the  vertex  be  the  origin  and  a  the  altitude. 

.        .      n  4-  3 
Ant.  m  =  — i— J  a. 

n  +  4 

44.  Find  the  centre  of  gravity  of  a  ccae,  the  density  of 
every  particle  of  whicli  increases  as  its  distance  from  the 


\a,  whore  the  vertex  is  the  origin  and  a  tha 


Ans.  w 
altitude. 


46.  Find  the  centre  of  gravity  of  the  volume  of  uniform 
density  ounlained  between  a  hemisphere  and  a  cone  whoso 
vertex  is  the  vertex  of  the  hemicphcre  and  base  is  the  bast) 
of  the  hemisphere. 


3?ji^«««f*«w»tsw,'aaH«»»MS««if«ii^^ 


3a  of  ft  qaadrant 

y  point  varies  as 

xis. 

■avity  of  the  sur- 

Ans.  i  =  ^a. 

rface  formed  by 

ins.  i  = 


50a 
63* 


find  the  centre 

;ween  the  vertex 

a  distance  from 

(latns  rectum). 

the  density  of 
th  power  of  its 
ertei. 
itude. 

n  +  3 


n  +  4 


a. 


the  density  of 
ituDce  from  the 

igin  and  a  r.he 

me  of  aniforni 
a  cone  whoso 
Hvse  is  the  baso 


XXAMPLSSL 


147 


a 


Ans.  *  =  2>  ^^^^  the  vertex  is  the  origin  and  a  the 
altitude. 

46.  Find  the  distance  of  the  centre  of  gravity  of  a  henii- 
aphere  from  the  centre,  the  radius  being  a, 

Ans.  i  =  |fl. 

47.  Find  the  centre  of  gravity  of  the  solid  generated  by 
the  revolution  of  the  cycloid, 

y  =  V2ax  —  a»  +  a  vers"*?, 

a 

(1)  round  the  axis  of  z,  and  (2)  round  the  axis  of  y. 

A        /i\-        (637r»_64)«      ....        /16       Tr»\  2a 

Am.  (1)  .  =  V(9^;ii)-5  (2) »  -  (y  +  i)  IT- 

48.  Find.the  centre  of  gravity  of  the  volume  formed  by 
the  revolution  round  the  axis  of  x  of  the  area  of  the  curve 


f  —  axf  +  X*  —  0. 


^ 


Ant.  i 


3an 
32  ' 


49.  Find  the  centre  of  gravity  of  the  volume  generated 
by  the  revolution  of  the  area  in  Ex.  2U  round  the  axis  of  y. 

.       -  8a 

^'"- >' =  aTr5inri4)- 

60.  Find  the  centre  of  gravity  of  a  hemisphere  when 
the  density  varies  as  the  square  of  the  distance  from  the 
ooutro.  .       .       5a 

''12' 


Ant. 


51.  Find  the  centre  of  gravity  of  the  solid  generated  by  a 
scmi*pai-abola  bounded  by  the  latns  reotum,  rsvolviug 
round  the  lutus  reotum. 

Ant.  Distance  from  focus  =  /^  of  latus  reotum.     ^ 


.  (I'fc'fi 


148 


SXAMPLBS. 


62.  The  vertex  of  a  right  circular  cone  is  at  the  centre  of 
a  sphere ;  find  the  centre  of  gr&vity  of  a  body  of  uniform 
density  contained  within  the  couo  and  the  sphere. 

Ans.  The  distance  of  the  centre  of  gravity  from  tue  ver- 

tex  of  the  cone  =  ~  (1  +  cos  «),  where  a  =  the  semi- 

o 

vertical  angle  of  the  cone  and  a  =  the  radius  of  the 
sphere. 

53.  Find  the  distance  firom  the  origin  to  the  centre  of 
gravity  of  the  solid  generated  by  the  revolution  of  the 
cardioid  round  its  prime  radius,  its  equation  being 

r  =  a  (1  +  cos  6). 

Ana.  X  =  fa. 

54.  Find  by  Art.  90  (1)  the  surface  and  (2)  the  volume 
of  the  solid  formed  by  the  revolution  of  n  cycloid  round 
the  taigent  at  its  vertex. 

Ans.  Surface  =  ^na';  Volume  =  rrV. 

55.  Find  (1)  the  surface  and  (2)  the  volume  of  the  solid 
formed  by  the  revolution  of  a  cycloid  round  its  liaso. 

Am.  (1)  *^na* ;  (2)  orV. 

56.  Au  equilateral  triangle  revolves  round  its  base, 
whose  length  is  a ;  find  (1)  the  area  of  tho  surface, 
and  (i)  the  volume  of  the  figure  described. 

Ana.  (1)  na*  a/3  ;  (2)  -^. 

67.  Find  (1)  the  surface  and  (2)  the  volnmo  of  a  ring 
with  a  circular  section  whose  intomal  diameter  is  12  ins., 
and  thickness  3  ins. 

Am.  (1)  444.1  sq.  in.;  (2)  833.1  cub.  in. 


at  the  centre  of 

•ody  of  uoifonu 

phere. 

y  from  tue  ver- 

a  =  the  Bemi- 

radiua  of  tbe 

0  the  centre  of 
relation  of  the 

1  heing 

ins.  X  =  |a. 

(2)  the  volume 
cycloid  round 

ume  =  tV. 

me  of  the  Rolid 

its  liasc. 

»» ;  (2)  5rW. 

und    its    base, 
tho   surface, 


\/3 ;  (2) 


iTrt» 


nniu  of  a  ring 
iieter  is  12  ins., 

33.1  cnb.  in. 


CHAPTER    V. 

FRICTION. 

91.  Friction. — Friction  is  that  force  which  acts  between 
two  bodies  at  their  surface  of  contact,  and  in  the  direction 
of  a  tangent  to  that  surface,  so  as  to  resist  their  sliding  on 
each  other.  It  depends  on  the  force  with  which  the  bodies 
are  pressed  together.  All  the  curves  and  surfaces  which  we 
have  hitherto  considered  were  supposed  to  be  smooth,  and, 
as  such,  to  offer  no  resistance  to  the  motion  of  a  body  in 
contact  with  them  in  any  other  than  a  rormal  dircctiou. 
tSuch  curves  and  surfaces,  however,  are  rot  to  be  tbund  in 
nature.  Every  surf  two  is  capable  of  destroying  a  certain 
amount  of  force  in  its  tangent  plane,  i.e.,  it  possesses  a  certain 
degree  of  roughness,  in  virtue  of  which  it  resists  tho  sliding 
of  other  surfaces  upon  it.  This  resistance  is  called  friction, 
and  is  of  two  kinds,  viz.,  sliding  and  rolling  friction.  The 
rst  is  that  of  a  heavy  body  dragged  on  a  plane  or  other 
surface,  an  vie  turning  in  a  fixed  box,  or  a  vertical  shaft 
turning  on  a  narizo'zU!  plate.  Friction  of  the  Be<,ond  kind 
is  that  of  a  v;beel  rolling  along  a  plane.  Both  kinds  of 
friction  are  governed  by  the  same  laws ;  the  former  is  much 
Teater  than  the  latter  under  the  same  circumstances,  and 
<    I  he  only  one  that  we  shall  consider. 

.'\.  smooth  surface  i  i  one  which  opposes  no  resistance  to 
the  motion  of  a  body  upon  it  A  rough  surface  is  one 
which  does  oppose  a  resistance  to  the  motion  of  a  body 
u])ou  it. 

TIiP  fliirfkcM  of  all  bodiM  eonslBt  of  rerj  Rintll  elevations  and 
deproMlons,  ho  that  if  thejr  are  prMsed  againut  vacli  (ith«r,  the 
elevatlona  of  ooo  fit,  mom  or  lesa,  into  the  deprpwions  of  thv  other, 
and  tbe  Burfaoea  iitt«rp«netrate  each  otkor ;  and  the-  mutual  penetra- 


\ht 


160 


LAWS  OF  FRIcnON. 


tion  is  of  ooane  greater,  if  the  presBing  force  is  greater.  Henoe, 
when  a  force  is  applied  bo  as  to  cause  one  liody  to  move  on  another 
with  which  it  is  iu  contact,  it  is  necessary,  lieforo  motion  can  take 
place,  either  to  break  off  the  elevations  or  compress  them,  or  force  tlio 
iKxliea  to  separate  far  enough  to  allow  them  to  pass  each  other. 
Much  of  this  roughneM  may  be  removed  by  polishing ;  and  the  effect 
of  much  of  it  may  be  destroyed  by  lubrication. 

Friction  always  vets  along  a  tangent  to  the  surfiuje  at  the  point  of 
contact ;  and  its  direction  is  opposite  to  that  of  the  lice  of  motion  ;  it 
presents  itself  in  the  motion  of  a  body  as  a  passive  force  or  resistance,* 
since  it  can  only  hinder  motion,  but  can  never  produce  or  aid  it.  In 
investigations  in  mechnnius  it  can  be  considered  as  a  force  acting  in 
opposition  to  erery  motion  whose  direction  lies  in  the  plane  of  contact 
of  the  two  bodies.  Whatever  may  be  the  direction  in  which  we  move 
a  body  resting  upon  a  boriaontal  or  inclined  plane,  the  friction  will 
always  act  in  the  opposite  direction  to  that  of  the  motion,  i. «.,  when 
we  slide  a  body  down  an  inclined  plane,  it  will  appear  as  a  force  up 
the  plane.  A  surface  may  also  resist  sliding  motion  by  means  of  tha 
adhetion  >  ^ween  its  substance  and  that  of  another  body  in  contact 
with  it.f 

Tho  friction  of  a  body  on  a  surface  is  measured  by  the 
leaxt  force  which  will  put  the  body  in  motion  along  the 
surface. 

92.  Laws  of  FriotioB. — In  onr  ignorance  of  the 
ooudtitutiun  of  bodies,  the  laws  of  friction  must  be  deduced 
from  experiment.  Experiments  made  by  Coulomb  and 
Morin  have  established  the  following  laws  of  friction : 

(1)  The  friction  varies  as  the  normal  pressure  when  the 
materials  of  the  surf..^js  in  contact  remain  the  same.  Subse- 
quent experiments  have,  however,  considerably  modified 
this  law,  and  shown  that  it  can  be  regarded  only  as  an 
approximation  to  the  truth.  When  the  pressure  is  very 
great  it  is  found  that  tho  friction  is  less  than  this  law 
would  give. 


*  WalHbwh,  p.  «». 

t  8e«  Haaklue'i  ApplM  MsetMDks,  p.  au». 


I  greater.  Henoe, 
0  movA  on  another 
3  motion  can  take 
them,  or  force  tlie 
paw  each  other, 
ag  i  and  the  efl'cxrt 

aoe  at  the  point  of 
line  of  motion  ;  it 
brce  or  resistance,* 
duee  or  aid  it.  In 
ts  a  force  acting  in 
lie  plane  of  contact 
in  which  we  move 
le,  the  friction  will 
motion,  i.e.,  when 
>pear  as  a  force  up 
a  by  means  of  tha 
ler  body  in  contact 


leasured  by  the 
)tion  along  the 


loranoe   of   the 
ust  be  deduced 
Coulomb  and 
frictiou : 

|e«jiur(?  wJien  the 
same,    Subsc- 

Irsbly  modified 
only  as  nn 
assure  is  very 
than  this  law 


ijiwa  or  rRicnoN. 


(2)  Tht  frictioH  is  independent  of  the  extent  of  the  aur- 
facee  in  contact  so  hng  as  the  normal  pressure  remains  the 
same.  When  the  surfacec  iu  contact  are  very  small,  as  for 
instance  a  cylinder  resting  on  a  surface,  this  law  gives  the 
friction  much  too  great. 

These  two  laws  are  true  when  the  body  is  on  the  point  of  moying, 
and  also  when  it  is  actually  in  motion ;  but  in  the  case  of  motion  the 
magnitude  of  the  friction  is  not  always  the  same  as  when  the  body  is 
beginning  to  move  ;  when  there  is  a  difference,  the  fHction  is  greater 
in  the  state  bordering  on  motion  than  In  actual  motion. 

(3)  The  friction  is  independent  of  the  velocity  when  the 
body  is  in  motion. 

It  follows  from  those  laws  that,  if  .R  be  the  normal 
pressure  between  the  bodies,  F  the  force  of  friction,  and  ft 
the  constant  ratio  of  the  latter  to  the  former  when  slipping 
is  about  to  ensite,  we  have 


F=  fiR. 


(1) 


The  fraction  n  is  called  the  co-efficient  qf friction  ;  and  if 
the  first  law  were  true,  fi  would  be  strictly  constant  for  the 
■ame  pair  of  bodies,  whatever  the  magnitude  of  tho  normal 
pressure  between  them  might  bo.  This,  however,  is  not 
tho  case.  When  the  normal  pressure  is  nearly  equal  to  that 
which  would  crush  either  of  the  surfaces  in  contact,  the 
force  of  friction  increases  more  rapidly  than  the  normal 
pressure.  Equation  (1)  is  nevertheless  very  nearly  true 
when  the  differences  of  normal  pressure  are  not  very  jfr<?at ; 
und  in  what  follows  we  shall  assume  this  to  be  the  case. 

Rbmabk. — The  laws  of  friction  were  established  by  Coulomb,  a 
distinguished  French  officer  of  Engineers,  and  were  founded  on 
expeHments  made  by  him  at  Rochefort.  The  results  of  these  experi- 
ments were  presented  in  1781  to  the  French  Academy  of  ScienreH,  and 
in  178A  his  Memoir  on  Friction  was  published.  A  very  full  nbHlract- 
of  this  paper  is  given  in  De  Young's  Natural  PhUoKtphy,  Vol.  II, 
p.  170  (1st  Ed.).  Further  ex|it>rinienM  wert;  made  at  Mi-ts  by  Morin, 
1881-S4,  by  direction  of  the  French  military  authorities,  the  raialt  of 


I  f . 


153 


AlfOLK  OF  FRICTION. 


i 


which  has  been  to  oonfirm,  with  slight  exeeptions,  all  the  results  of 
Coulomb,  and  to  determine  witli  conBldersble  precision  tlie  numerical 
values  of  the  coefficients  of  friction,  for  all  the  substances  usually 
employed  in  the  construction  of  machiaes.  (8ee  Oalbraith's  Me- 
chanics, p.  68,  Tiwisdeu's  Practical  Mechanics,  p.  188,  and  Weisbach's 
Mecbauica.  VoL  I,  p.  817.) 

93.  Magnitndes  of  CoeffloientB  of  Friction.— Prac- 
tically there  is  no  observed  coeflRcient  much  greater  than  1. 
Most  of  the  ordinary  coefficients  are  less  than  ^.  The  fol- 
lowing results,  selected  from  a  table  of  coefficients,*  will 
afford  an  idea  of  the  amount  of  friction  as  determined  by 
experiment :  these  results  apply  to  the  friction  of  motion. 

For  iron  on  stone      i*  varies  between  .3  and  .7. 
For  timber  on  timber "      "  "       .2  and  .5. 

For  timber  on  metals  "      "  "       .2  and  .6, 

For  metals  on  metals  "      "  '*      .15  and  .25. 

For  full  particulars  on  this  subject  the  student  is  referred 
to  Rankine's  Applied  Mechanics,  p.  209,  and  Mosilcy's 
Engineering,  p.  124,  also  to  the  treatise  of  M.  Morin,  where 
he  will  find  the  subject  investigated  in  all  its  completeness. 

94.  Angle  of  Friction. — The  angle  at  which  a  rough 
plane  or  surface  may  be  inclined  so  that  a  body,  when  acted 
upon  by  the  force  of  gravity  only  may  just  rest  upon  it  with- 
out sliding,  is  called  the  Angle  of  Friction,  f 

Let  a  bo  the  angle  of  inclination  of 
the  piano  AB  just  as  the  weight  is  on 
the  point  of  slipping  down;  W  the 
weight  of  the  body ;  R  the  normal  pres- 
sure on  the  plane ;  F  the  force  of  fric- 
tion acting  along  the  plane  =  fiR  (Art. 
92).  Then,  resolving  the  forces  along  and  perpendicular  to 
the  plane  we  have  for  equilibrium 

*  Ranklne'B  AppHod  Mechautcs,  p.  m. 

t  SoiuetimM  called  "  Ibe  aogto  of  ropoae;  "  al»o  caUad  "  the  Umltlng  asglo  of 
resUtaiMe." 


na-45 


all  the  rasolts  of 
ion  tlie  namerical 
ubatauces  nsnalljr 

Galbnith's  Me- 
B,  and  Weisbach's 

fiction.— Prac- 

greater  than  1. 
an  f  The  fol- 
efficiente,*  will 
determined  by 
on  of  motion. 

.3  and  .7. 

.2  and  .5. 

.2  and  .6. 

15  and  .25. 

dent  is  referred 
and   Mosiley'fl 
Morin,  where 
)  completeness. 

which  a  rough 
ody,  tvhen  acted 
upon  it  with' 


erpendicnlar  to 


the  limiting  angle  of 


BK ACTION  or  A  SOUOH  CVBVK. 


uB  =  ffaina;  R  —  ffoosa; 


tan  a  =  /», 


a) 


which  gives  the  limiting  valae  of  the  inclination  of  the 
plane  for  which  equilibrium  is  possible.  The  body  will  rest 
on  the  plane  when  the  angle  of  inclination  is  less  than  the 
angle  of  friction,  and  will  slide  if  the  angle  of  inclination 
exceeds  that  angle ;  and  this  will  be  the  case  however  great 
W  may  be ;  the  reason  being  that  in  whatever  manner 
we  increase  W,  in  the  same  proportion  we  increase  the 
friction  upon  the  plane,  which  serves  to  prevent  IT  from 
sliding. 

From  (1)  we  see  that  the  tangent  of  the  angle  of  friction 
is  equal  to  the  coefficient  of  friction. 

95.  ReactiOB  of  a  Rough  Cnrvo 
or  Surface. — Let  AB  be  a  rough  cuive 
or  surface ;  P  the  position  of  a  particle 
on  it ;  and  suppose  the  forces  acting  on 
P  to  be  confined  to  the  plane  of  the 
paper.  Let  R^  =  the  normal  resistance  of  the  surface, 
acting  in  the  normal,  PN,  and  F  =  the  force  of  friction, 
acting  along  the  tangent,  PT. 

The  resultant  of  if,  and  F,  called  the  Totai  Resistance* 
of  the  surface,  is  represented  in  magnitude  and  direction  by 
the  line  PR  =  R,  which  is  the  diagonal  of  the  parallelo- 
gram determined  by  R^  and  F.  We  have  seen  that  the 
total  resistance  of  a  smooth  surface  is  normal  (Art.  41) ;  but 
this  limitation  does  not  apply  to  a  rough  surface.  Let  ^ 
denote  the  angle  between  R  and  the  normal  £, ;  then  ^  is 
given  by  the  equation 

*  tan  0  =  ^• 

*  MlBcitia'p  Btatici,  p.  (M. 


ris.4e 


154 


WBiCTION  ON  Air  INCLiySD  PLAKIl, 


Hence,  ^  will  be  a  maximum  when  the  force  of  friction, 
F,  bears  the  greatest  ratio  to  the  normal  pressnre  B^.  But 
this  greatest  ratio  is  attained  when  the  body  is  just  on  the 
point  of  slippin/;  along  the  surface,  and  is  what  we  called 
the  coefficient  of  friction  (Art  92),  that  is 


Tr  =  ^; 


.  • .    t«m  ^  =  ^. 

Therefore  the  greatest  angle  by  which  the  Total  Reaietanc^ 
of  a  rough  curve  or  surface  can  deviate  from  the  normal  is 
the  angle  whose  tangent  is  the  coefficient  of  friction  for  the 
bodies  in  contact ;  and  this  deviation  is  attained  when  slip- 
ping is  about  to  commence. 

Cob.— By  (1)  of  Art  94,  tan  «  =  f» ; 


ft. 


hence,  the  direction  of  the  total  resistance,  R,  is  inclined  at 
an  angle  a  to  the  normal ;  i.e.,  the  greatest  angle  that  the 
Total  Resistance  of  a  rough  curve  or  surface  can  make  with 
the  normal  is  equal  to  the  angle  of  friction,  correspmiding 
to  the  two  bodies  in  contact. 

96.  Friction  on  an  Inclined  Plane. — A  body  rests  on 
a  rough  inclined  plane,  and  is  acted  on  by  a  given  force,  P, 
in  a  vertical  plane  which  is  perpendicular  to  the  inclined 
plane ;  find  the  limits  of  the  force,  and  the  angle  at  which 
the  least  force  capable  of  drawing  the  particle  up  the  plane 
must  act 

Let  t  =  the  inclination  of  the  plane  to  the  horiison  ;  6  = 
the  angle  between  the  inclined  plane  and  the  line  of  action 
of  P;  (I  =■  the  coefficient  of  friction ;  aud  let  us  first  snp- 
pose  that  the  body  is  on  the  point  of  moving  down  the 


force  of  friction, 
)Tes8are  B^.  But 
»dy  is  just  on  the 
is  ivhat  we  called 


i  Total  Betistance' 
om  the  normal  is 
''  friction  for  the 
tained  when  slip- 


R,  is  inclined  at 
st  angle  that  the 
ce  can  make  with 
m,  correspotuling 


A  body  rests  on 
a  given  force,  P, 
to  the  inclined 
angle  at  which 
cle  up  the  plane 

le  horizon ;  0  = 

le  line  of  action 

let  us  first  snp- 

loving  dowti  the 


mmmmmmmBm^':. 


FRICTION  ON  AN  INCLINMD  PLANS. 


156 


plane,  so  that  friction  is  a  force  acting  up  the  plane^  then 
resolving  along,  and  perpendicular  to,  the  plane,  we  have 

F  +  Pco&e  =z  W fan  if 

B  +  Pmie=  Wooai, 

F^fiB; 

.  sin  t  —  ^  cos  t 


P=W 


cobO  —  fianO 


(1) 


And  if  P  is  increased  so  that  motion  up  the  plane  is  just 
beginning,  F  acu  in  an  opposite  direction,  and  therefore 
the  sign  of  /u  must  be  changed  and  we  have 


p  —   j|r»^P*'4-^C0St 

COS  $  +  n  Bind' 


(2) 


Hence,  there  will  be  equilibrium  if  the  body  be  acted  on  by 
a  force,  the  magnitude  of  which  lies  between  the  values  of 
P  in  (1)  and  (2).  Substituting  tan  ^  for  /«  (Art.  95) ;  (2) 
becomes 


n_  nrwn(t-f  ») 


COB 


(«) 


To .  determine  6  in  (2)  so  that  P  shall  be  a  minimum  we 
must  put  the  first  derivative  of  P  with  respect  to  fl  =  0, 
therefore 

dP       nr  /  •     •  ■  ^    and  — fi  COB  6 

,*.    tan  9=:^; 

that  is,  the' force  P  necessary  to  draw  the  body  up  the  plane 
will  be  the  least  possible  when  0  =.  the  angle  of  friction. 


166 


DODBhE-UrCLINMD  PLANE. 


Henoe  we  infer  that  a  given  force  acts  to  the  greatest 
advantage  in  dragging  a  weight  up  a  hill,  if  the  angle  ut 
which  its  line  of  action  is  inclined  to  the  hill  is  equal  to 
the  angle  of  friction  of  the  hilL  Similarly,  a  force  acts  to 
the  greatest  advantage  in  dragging  a  weight  along  a  hori- 
zoatal  plane  if  its  line  of  action  is  inclined  to  the  plane 
at  the  angle  of  friction  of  the  plane.  We  may  also  deter- 
mine from  this  the  angle  at  which  the  traces  of  a  drawing 
horse  should  be  inclined  to  the  plane  of  traction. 

These  results  are  those  which  are  to  be  expected,  becanse 
some  part,  of  the  force  ought  to  be  expended  in  lifting  the 

ight  from  the  plane,  so  that  friction  may  be  diminished. 
V  oe  Price's  Anal.  Mech's,  Vol.  I,  p.  160.) 


97.  Friction  on  a  Doubla-Inclined  Plane.— Two 
bodies,  whose  weights  are  P  and  Q,  rest  on  a  rough  double- 
inclined  plane,  and  are  connected  by  a  string  which  passes 
over  a  smooth  peg  at  a  point.  A,  vertically  over  the  intersec- 
tion, fi,  of  the  two  planes.  Find  the  position  of  equili- 
brium. 

Let  «  and  /3  be  the  inclinations  of 
the  two  planer  ;  let  2  =  the  length  of 
the  string,  and  h  =  AB;  and  let  0 
and  0'  be  the  angles  the  portions  of 
the  string  make  with  the  planes. 

Suppose  P  is  on  the  point  of 
ascending,  and  Q  of  descending. 
Then,  since  the  motion  of  each  Inxiy  is  about  to  ensue,  the 
total  resistances,  R  and  ^S',  must  each  make  the  angle  of 
friction  with  the  corresponding  normal  (Art  95,  Oor.) ;  and 
since  the  weight,  P,  is  about  to  move  upwards  the  friction 
must  act  downwards,  and  therefore  R  must  lie  below  the 
normal,  while,  since  Q  is  about  to  move  downwards,  the 
friction  must  act  upwards,  and  therefore  8  must  be  above 
the  normal. 


Fig.4y 


&ilKiS^&- 


i  to  the  greatest 
,  if  the  angle  at 
)  hill  is  equal  to 

a  force  acts  to 
it  along  a  hori- 
led  to  the  plane 

may  also  deter- 
ces  of  a  drawing 
;tion. 

tpeeted,  because 
;d  in  lifting  the 
'  be  diminished. 


1  Plana.— Two 

a  rough  double- 
g  wliich  passes 
ver  theintersec- 
ition  of  equili< 


F(s.47 

i  to  ensue,  the 
e  the  angle  of 
95,  Oor.) ;  and 
ds  the  friction 
t  lie  below  the 
ownwards,  the 
must  be  above 


DOUBLE-mcUNSD  PLAIflS. 


187 


If  T  is  the  tension  of  the  string,  we  hare  for  the  eqni- 
libriam  of  P,  (Art.  32), 

y^j>  «»(«  +  ») 
cos  (d  —  1^) 

And  for  the  equilibrium  of  Q, 

~  ^co8(e'  ■\-<p)' 

Equating  the  v^ues  of  ^  we  get 

pSin(«-|-^)  _      sin  (0-0) 
C08((?  — 0)        ^co8(e'4-<>)' 

and  if  P  is  about  to  move  down  the  piano,  the  friction  acts 
in  an  opposite  direction,  and  therefore  the  sign  of  0  must 
be  changed  and  we  have 


„  sin  («  —  0)  _     sin  (/?  +  0) 
cos  {0  +  <p)       ^  cos  (»'  —  0)' 


(2) 


(1)  or  (2)  is  the  only  statical  equation  connecting  the 
given  quantities. 

We  obtain  a  geometric  equation  by  expressing  the  length 
of  the  string  in  terms  of  h,  «,  /3,  6,  and  B',  which  is 


/ 


Vsin  e  ^  sin  Bf 


(3) 


From  (1)  or  (2)  and  (3)  the  values  of  6  and  0'  can  be  found, 
and  this  determines  the  positions  of  P  and  Q. 

Otherwise  thus : 

Instead  of  considering  the  total  resistances,  R  and  S,  we 
may  consider  two  iwrmai  resistances,  Ji^  and  Si,  and  two 


158 


DOUBLE-INCLISSD  PLANK, 


forces  of  friction,  /»iJ,  and  /u/S,,  acting  respectively  down 
the  plane  «  and  up  the  plane  /3.  In  this  case,  considering 
the  equilibriam  of  P,  and  resolving  forces  along,  and  per- 
pendicular to,  the  plane  a,  we  have 


P  sines  +  M^,  =  TcmB, 
P  cos  a  =  i?i  +  T  sin  e, 


:■] 


(4) 


and  for  the  eqnilibriam  of  Q, 


C8in/3  =  |ii/S,  +  TcosffA 
Cco8/3=  /S,  +  Tsmff.    I 


(5) 


Eliminating  i?,,  S^,  and  T  from  (4)  and  (5)  we  get  (1), 
the  same  statical  equation  as  before. 

The  method  of  considering  total  resistances  instead  of 
their  normal  and  tangential  components  is  usually  more 
simple  than  the  sep8,rate  consideration  of  the  latter  forces. 
(See  Minchin's  Statics,  p.  60.) 


CoE. — If  Q  is  given  and  P  be  so  small  that  it  is  abont  to 
ascend,  its  value,  P^,  will  be  given  by  (1), 

P    _  ft  ^inj^^^l^ii?:!!^  ,R^ 

^^-^  flfn  (a  +  ^)  COS  {ff  +  0)'  ^' 

nnd  if  P  is  so  large  that  it  is  abont  to  drag  Q  up,  its  value, 
P,,  will  be  given  by  (2) 


P,  =  Q 


sin  (/3  +  «f>)  COB  (0  +  ^) 
sin  (o  —  0)  cos  ((?'  —  0) 


(7) 


the  angles  0  and  6'  being  connected  by  (3). 

There  will  be  equilibrium  if  ^  be  acted  on  by  any  force 
whose  magnitude  lies  between  P|  and  Pg. 


fc — 


espectively  down 

case,  considering 

along,  and  per- 


(*) 


(6) 


i  (5)  we  get  (1), 

ances  instead  of 
is  usually  more 
;he  latter  forces. 


lat  it  is  about  to 

ho"  <*^> 

Q  up,  its  value, 
n  by  any  force 


KiiiiaiiilliMWIMIWipWIWI'l 


fmm 


ut'iiimnmHmmimmMmixiKn' 


FRICTION  OF  A    TRUNNIOie. 


159 


Pi8.4S 


98.  FrictioB  on  Two  XncUned  Planes.— A  beam 
rt'sts  ou  two  rough  inclined  planes;  find  the  position  of 
equilibrium. 

Let  a  and  b  be  the  segments,  AG 
and  BG,  of  the  beam ;  let  0  be  the 
inclination  of  the  beam  to  the  hori- 
zon, a  and  /3  the  inclinations  of  the 
planes,  and  R  and  S  the  total  resist- 
ances. Suppose  that  A  is  on  the 
point  of  ascending;  then  the  total 
resistances,  R  and  8,  must  each 
make  the  angle  of  friction  with  the  correeponding  normal 
and  act  to  the  right  of  the  normal. 

The  three  forces,  W,  R,  S,  must  meet  in  a  point  0  (Art. 
62) ;  and  the  angles  GOA  and  GOB  are  equal  to  a  -f-  0, 
and  (3  —  (p,  respectively. 

Hence    {a  +  b)  cot  BGO  =  o  cot  GOA  —  b  cot  GOB, 

or      (a  +  ft)  tan  e  =  o  cot  (a  +  0)  —  i  cot  (/3  —  0).     (1) 

Cob. — If  the  planes  are  smooth,  ^  =  0,  and  (1)  becomes 

{a  +  b)  taxi6  =  acota  —  b  cot  j3. 
(See  Ex.  7,  Art  62.) 

99.  Friction  of  a  Trunnion.* — Trunnions  are  the 
cylindrical  projections  from  the  ends  of  a  shaft,  which  rest 
on  the  concave  surfaces  of  cylindrical  boxes.  A  shaft  rcEits 
in  a  horizontal  position,  with  its  trunnions  ou  rough 
cylindrical  surfaces;  find  the  resistance  due  to  friction 
which  is  to  be  overcome  when  the  shaft  begins  to  turn 
about  a  horizontal  axis. 

«  BometimM  called  "  JonmaL" 


160 


mWTrON  OP  A   PIVOT. 


FiR.4» 


Let  IM  and  BAED  be  two  right 
flections  of  the  trunnion  and  its  box; 
the  two  circles  are  tangent  to  each 
other  internally.  If  no  rotation 
takes  place  the  trunnion  presses 
upon  its  lowest  point,  H',  through 
which  the  direction  of  the  resulting 
pressure,  R,  passes ;  if  the  shaft 
begins  to  rotate  in  tiie  direction  AH,  the  trunnion  ascends 
along  the  inclined  surface,  EAB,  in  consequence  of  the 
friction  on  its  bearing,  until  the  force,  ^S,  tending  to  move 
it  down  just  balances  the  friction,  /'.  Resolving  R  into  a 
normal  force  AT  and  a  tangential  one,  8,  we  have,  since  the 
t<<ngential  coini>onent  of  R  in  urging  the  trunnion  down 
the  burface  =  tlie  Mction  which  opposes  it. 

•8=F-(iN;    but    R*  ^  S*  +  N*; 


or 

therefore 

and  the  friction 


R*  =  ix*2P  +  I^; 

R 


N- 


fiR 


Vi  +^«' 


Rtantft 


or 


vT-f  n*       Vl  +  tanS  0 
P  =r  Ram  (p. 


(Art.  96), 


Ilonco,  fo  fiiv^  (he  friction  upon  n  (rtinnioti,  muUiply  the 
resultant  of  the  forces  which  act  upon  it  by  the  sine  of  the 
angle  of  friction. 

100.  Friction  of  •  Pivot—A  heavy  circular  shaft 
rests  in  a  vortical  position,  with  its  end,  which  is  a  circular 


Fip.49 


trunnion  ascends 
isequcnce  of  tho 
,  tending  to  move 
aohing  R  into  a 
e  have,  since  the 
>o  traimion  down 
t. 


+  N*', 


rt.  95), 


)//,  multiply  the 
y  the  sine  of  the 


circular  shaft 
til  is  a  circular 


SSfi^^ 


FBicnoN  Of  A  prroT. 


IGI 


section,  on  a  horisontal  plate;  And  tho  resigtimoe  due  to 
friction  which  is  to  be  overcon^e,  when  tho  shaft  begins  to 
revolve  about  a  vertical  axis. 

Let  a  be  the  radius  of  the  circular  section  of  the  shaft; 
let  the  plane  of  ( •.  $)  be  the  horizontal  one  of  contact 
between  the  end  of  the  shaft  and  the  plate;  and  let  tho 
centre  of  the  circular  area  of  contact  be  the  pole.  Let 
W  =  the  weight  of  the  siiaft,  then  the  vertical  pressure  on 

W 

each  unit  of  surface  is  — j;  and  therefore,  if  rdrd6  is  the 

area-element,  we  have 

the  pressure  on  the  element 


W 


rra- 


-,  r  dr  de  ; 


the  friction  of  the  element  sz  n~-  rdr  dO. 


no' 


The  friction  is  opposed  to  motion,  and  the  diroc-  jii  oi  its 
action  is  tangent  to  the  circle  descril)ed  by  the  element ; 
the  moment  of  the  friction  about  the  vertical  axis  through 
the  centre 

^Wt^drde 
■^  "~        ita*"  ' 

therefore  the  moment  of  friction  of  the  whole  circular  end 


«/o     t/fl 


(iWr^drde 


2ftWa 
8     ' 


(1) 


and  consequently  varies  aa  the  radius.  Hence  arises  the 
advantage  of  reducing  t«  tho  smallest  possible  dimensions 
llio  area  of  the  ba«o  of  a  vertical  shaft  revolving  with  its 
end  reiiting  on  a  horizontal  be<l. 

From  (1)  wn  -nay  regard  the  whole  friction  due  to  the 
pressure  oa  acting  at  a  single  point,  and  at  a  distance  from 
the  centre  of  motion  equ«l  to  two-thirds  of  the  nulins  o( 


m 


BXAMPLES. 


the  base  of  the  shaft, 
lever  of  fi-iction. 


This  distance  is  called  the  mean 


; '  r 


When  the  shaft  is  veriicai,  and  rests  upon  its  circular  end 
in  a  cylindrical  socket  the  cylindrical  projection  is  called  a 
Pivot, 

EXAMPLES. 

1.  A  mass  whose  weight  is  750  lbs.  rests  on  a  horizontal 
plane,  and  is  palled  by  a  force,  P,  whose  direction  mal  es 
an  angle  of  15"  with  the  horizon  ;  determine  P  and  the 
total  resistance,  B,  the  coefficient  of  friction  being  .62. 

Ans.  P  =  413.3  lbs.;  iJ  =  756-9  lbs. 

2.  i>  lermine  P  in  the  last  example  if  its  direction  is 
hoiizontal.  Ans.  P  =  465  lbs. 

8.  find  the  force  along  the  piano  required  to  draw  a 
weight  of  25  tons  np  a  rough  inclined  plutte,  the  coefficient 
of  friction  being  -j^,  and  the  inclination  uf  the  plnno  being 
snch  that  7  tons  acting  along  the  plane  would  siip|)ort  the 
weight  if  the  plane  were  smooth. 

Ans.  Any  force  greater  than  17  tons. 

4.  Find  I'lo  force  in  the  preceding  example,  supposing 
it  to  act  at  the  most  advantageous  inclination  to  the  plaice. 

Ans.  ]5t^  tons. 

5.  A  ladder  inr-Hned  at  an  angle  of  60°  to  the  horizon 
rests  between  a  rouyh  pavement  and  the  smooth  wall  of  a 
house.  Show  that  if  the  ladder  kgin  to  slide  when  a  man 
hAB  osoopdod  so  that  his  centre  of  gravity  in  half  way  up, 
then  the  coefficient  of  friction  between  the  foot  of  the 
ladder  and  the  pavement  is  \  VS. 

6.  A  body  whose  weight  is  20  lbs.  ip  just  sustained  on  a 
rough  inclined  plane  by  a  horizontal  force  of  2  iba.,  and  a 
force  of  10  lbs.  along  the  piano  ;  the  coefficient  of  friction  is 
I ;  find  the  inclination  uf  thu  plane.        Ans.  tan~'  (||). 


lalled  the  mean 

iis  circular  end 
Hon  is  called  a 


on  a  horizontal 
iircctiou  mal  es 
dine  P  and  the 
I  being  .62. 
=  756-9  lbs. 

its  direction  is 
P  =  465  lbs. 

irod  to  draw  a 
!,  the  coefficient 
the  plnne  l)eing 
Id  support  the 

Lhau  17  tons. 

i])Ic,  supposing 
to  the  plaiiC. 
}5-j^  tons. 

to  the  horizon 

rnootfi  wall  of  ii 

c  when  a  mun 

is  half  way  up, 

foot  of  the 


sustained  on  a 
1  2  lt)0.,  and  ti 
it  of  frictioa  is 
tan-'(il). 


i-'mmmmmmm 


BXAMPLBS. 


163 


7.  A  heavy  body  is  placed  on  a  rough  pljine  whose 
inclination  to  the  horizon  is  sin  "*  (f),  and  is  counected  by 
a  string  passing  over  a  smooth  pulley  with  a  body  of  equal 
weight,  which  hangs  freely.  Supiwsing  that  motion  is  on 
the  point  of  t-nsu'ng  up  the  plane,  find  the  inclinatiop  of 
the  striag  to  the  plane,  the  coeCaoient  of  friction  being  \. 

Ana.  ^  =  2  tan"'  (i). 

8.  A  heavy  body,  acted  uiwn  by  a  force  equal  in  magni- 
tude to  its  weight,  is  just  about  to  ascend  a  rough  inclined 
plane  under  the  influence  of  this  force ;  find  the  inclination, 
d,  of  the  force  to  the  inclined  plane. 

Am.  e  =  ^  —  *■»  or  2(;fr  +  t  -  1  where  t  =  inclination 
2  * 

of  the  plane,  and  <>  =  angle  of  friction,    {d  is  here  sup- 
posed to  be  measured  from  the  upper  side  of  the  inclined 

plane).    If  5  >  2^  -f  »,  *  is  negative  and  the  applied  force 

will  act  towards  the  under  side. 

9.  In  the  first  solution  of  the  last  example,  what  is  the 
magnitude  of  the  pressure  on  the  plane  ? 

Ana.  Zero,    Explain  this. 

10.  If  the  shaft,  (Art.  100),  is  a  square  prism  of  the 
weight  W,  and  rotates  about  an  axis  in  its  centre,  prove 
that  the  moment  of  the  friction  of  the  square  end  varies  as 
the  side  of  the  square. 

11.  If  the  shaft  is  composed  of  two  equal  circular 
cylinders  placed  side  by  side,  and  rotatcH  about  the  line  of 
contact  of  the  two  cylinders,  show  that  the  moment  of  the 
friction  of  the  surface  in  contact  with  the  horizontal  piano 
_  3a/mlF 

-~      9n     ' 

12.  What  is  the  least  coefficient  of  friction  that  will 
allow  of  a  heavy  body's  being  just  kept  from  sliding  down 


EXAMPLES. 

an  inclined  piano  of  given  inclination,  the  body  (whoso 
weight  is  W)  being  sustained  by  a  given  horizontal  force,  P  ? 

An,,  l^'^P. 
»K  +  /'  tan  i 

13.  It  is  obsoi-ved  that  a  body  whose  weight  is  known  to 
be  fKf'flin  bo  juet  sustained  on  a  rough  inclined  plane  by  a 
liorizontil  force  P,  and  tlmt  it  can  also  be  just  sustained  on 
the  same  plane  by  a  force  Q  up  the  plane;  express  the 
angle  of  friction  in  terms  of  these  known  forces. 

Am.  Angle  of  friction  =  cos~* • 

14.  It  is  o'  ved  that  a  force,  ^,,  acting  up  a  rough 
inclined  pL  .11  ju«t  sustain  on  it  a  body  of  weight  W, 
and  that  a  for-^e,  Q^,  acting  up  the  piano  will  jest  drag  tho 
same  body  up ;  find  the  angle  of  friction. 

Ans.  Angle  of  friction  —  sin"' ^l-T-jiJ 

15.  A  hea>>  uniform  rod  rests  with  its  extremities  on 
the  interior  of  a  rough  vortical  circle;  find  the  limiting 
position  of  equilibrium. 

Ans.  If  2«  is  the  angh  subtended  at  tho  centre  by  the 
rod,  and  A  the  angle  of  friction,  the  limiting  inclination  of 
the  rod  to  the  horizon  is  given  by  the  equation 


tan  6 


sin  2A 


cos  2A  -f-  cob  2a 


16.  A  solid  triangular  prism  is  placed,  with  its  axia 
horiisonUil,  on  a  rough  inclined  i)lane,  the  inclination  of 
which  is  gradually  increased  ;  determine  tho  nature  of  the 
kititil  motion  of  the  prism. 

Am.  If  the  ta-ianglc  ABO  is  the  section  iwrpondionlar  to 
the  uxis,  aud  the  gide  AR  is  isi  contact  with  the  plane,  A 


B  body  (whoso 
zontal  force,  P  ? 
rtan  i  —  p 
V  +  .Tii^' 

^ht  is  known  to 
led  plane  by  a 
i8t  sustained  on 
s;  express  the 
ces. 
PW 

ig  up  tt  rough 

7  of  weight  W, 

jest  drag  tho 


xtreniities  on 
the  limiting 

centre  by  the 
inclination  of 


ith  its  axis 
U'linution  of 
atiire  of  the 


pudionlar 


he  plane, 


EXAMPLES. 


165 


being  the  lower  vertex,  the  initial  motion  will  be  one  of 
tumbling  if 

5»  +  3<?  -  o« 


l^> 


4A 


the  sides  of  the  triangle  being  a,  b,  c,  and  its  area  A.  If  ft 
is  less  than  this  value,  the  initial  motion  will  be  one  of 
slipping. 

17.  A  frustnaa  of  a  solid  right  cone  is  placed  with  its 
base  on  a  rough  inclined  plane,  the  inclination  of  which 
is  gradually  increased ;  determine  the  nature  uf  the  initial 
motion  of  the  body. 

Am.  If  the  radii  of  the  larger  and  smaller  scctionB  arc  R 
and  r,  and  h  is  the  height  of  the  frustum,  the  initial  motion 
will  be  one  of  tumbling  or  slipping  according  as 


'*><T 


4R     R*  +  Rr  +  t* 


R*  +  2Rr  +  3r» 


18.  An  elliptic  cylinder  rests  in  limiting  equilibrium 
between  a  ro'igh  vertical  and  an  equally  rough  horizontal 
plane,  the  nxis  of  the  cylinder  being  horizontal,  and  the 
major  axis  of  the  ellipse  inclined  to  the  horizon  at  an  angle 
of  46°.    Find  the  coefficient  of  friction. 

Ans.  ft  =  —  ■  ■  -  _"~ — ^-,  e  being  the  eccentricity 
of  the  ellipse. 


P 


ffifiiiinn 


CHAPTER    VI. 


THE    PRINCIPLE    OF    VIRTUAL   VELOCITIES* 

101.  Virtual  Velocity.— If  the  point  of  application  of 
a  force  be  conceived  as  displaced  through  an  indefinitely  small 
spacs.  tho  resolved  part  of  the  displaceiHcnt  in  the  direction 
0  theforct^,  is  calkd  the  Virtual  Velocity  of  the  force,'  and 
the  product  of  the  force  into  the  virtual  velocity  has  been 
called  the  virtual  moment\  of  the  force. 

"J  hue,  let  O  be  the  original,  and  A 
the  new  point  of  application  of  the 
force,  i*,  acting  in  the  direction  OP, 
and  let  AN  be  drawn  perpendicular  to 
it.  Then  ON  is  the  virtual  relocity  of 
P,  and  P  •  ON  is  the  virtual  moment, 
virtual  displacement  of  the  point. 

If  the  projection  of  the  virtual  displacement  on  the  line 
of  the  force  lies  on  the  side  of  O  toward  which  P  acts,  as  in 
the  figure,  the  virtual  velocity  is  considered  positive;  but 
if  it  lies  </fi  the  opposite  side,  i.  «.,  on  the  action  line  pro- 
longed through  0,  it  is  negative.  The  forces  are  always 
n>garfied  as  positive ;  the  sign,  therefore,  of  a  virtual  mo- 
ment will  be  the  same  as  that  of  tho  virtual  v«'locity. 

Cob. — If  d  be  the  angle  between  the  force  and  the  virtual 
dibpiacement,  we  have  for  the  virti?al  moment, 

P .  ON  =  P  •  OA  cos  «  =  P  co£  e .  OA. 


OA  is  called  the 


*  Thf-  rHw)tpN>  3t  Vinnil  VeloctUwi  ww  dlfoovered  by  Otllleo,  mi  wm  rmj 
Ailly  d'-"    ''ped  by  BnrtiouiUi  and  La^gnngt. 


t  Soiiu  tlniM  oatM  ' 
bj  UuluuotJ. 


VirtiuU  Work."    Tte  ovaa  "  Virtiul  Momaat "  < 


i^ren 


.'JiM**» 


^■^ 


OCITIES* 

f  application  of 
idefinitely  small 
n  the  direction 
the  force ;  and 
tlocity  has  been 


Fig.50 

.  is  called  the 

it  on  the  line 
P  acts,  ofi  in 
positive;  but 
tion  line  pro- 
08  are  always 
a  virtual  mo- 
looity. 

nd  the  virtual 


OA. 

lleo,  and  ww  very 
omont "  WM  giTen 


'■illWgff'WIIIIIIIIIIIIIWWIUIlllll 


VIRfUAL    VELOCITIES. 


107 


Now  P  cos  6  is  the  projection  of  the  force  on  the  direction 
of  the  displacement,  and  is  equal  to  OM,  OP  being  the 
force  and  PM  being  drawn  perpendicular  to  OA.  Hence 
we  may  also  define  the  virtual  moment  of  a  force  as  the 
product  of  the  virtual  displacement  of  its  point  of  applica- 
tion into  the  projection  of  the  force  on  the  direction  of  this 
displacement;  and  this  definition  for  some  purposes  is 
more  convenient  than  the  former. 

Rbmahk.— A  forca  is  said  to  do  work  if  It  movM  the  body  to  wbidi 
it  is  applied ;  and  tlie  work  doDu  by  it  is  measuriMi  by  tlie  product  i>{ 
tlie  force  into  tiie  space  through  wliich  it  moTce  the  body.  Generally, 
the  work  done  by  any  force  during  an  infinitely  tuBall  dkaplacemeut  uf 
its  point  of  application  is  tlie  product  of  the  rettolved  paii  of  tbc  force 
in  the  direction  of  the  displacement  into  tlie  displaomnent .  hmI  thia 
is  the  same  as  the  virtutM^  «»>«M«Mt  of  Um  force. 


102.   Principle   of  ▼ht—l   "W^HocStx^m.  —  (1)    The 

virtual  moment  of  a  force  m  e'^ttui  iv  the  xum  of  the  virtual 
moments  of  its  components. 

Let  OR  represent  a  force,  ft,  act- 
ing at  0,  and  let  its  components  be 
/'  and  Q,  represented  by  OP  and 
OQ.  Let  OA  be  the  virtual  dis- 
placement of  0,  and  let  its  projec- 
tions on  R,  P,  and  Q,  be  r,  p,  and 
q,  respectively.  Then  the  virtual 
moments  of  these  forces  are  R  •  r,  P 
Pm,  and  Qo,  perpendicalar  to  OA. 


Fia.M 


p,  Q  •  q.  Draw  /?>/, 
Then  On,  Om,  and  Oo 
(=  mn),  are  the  projections  of  R,  P,  and  Q,  on  the  dirr-c- 
tion  of  the  displacement ;  and  henoe  (Art.  lOX,  Cor.)  we  have 

i2  .  r  =  OA  On ; 
P.;>=OA.Om; 
^  -  q  ss  OA  •  mn. 


168 


rinvuAL  VELoctTtsa. 


Hence  P  •  ^  +  Q  •  y  =  OA  (Om  +  mn) 

=  OA  '  On  =  E-r. 
(See  MinohiD'a  Statics,  p.  68.) 

(2)  If  there  are  any  number  of  component  forces  we  may 
compouud  them  iu  order,  taking  any  two  of  them  first,  and 
finding  the  virtual  moment  of  their  resultant  as  above,  then 
finding  the  virtual  moment  of  the  resultant  of  these  two 
and  a  third,  likewise  the  virtual  moment  of  the  resi!i>«nt  of 
the  first  three  and  a  fourth,  and  bo  on  to  the  last ;  or  we 
may  use  the  polygon  of  forcew  (Ai-t.  33).  The  sum  of  the 
virtual  moments  of  the  forces  is  equal  to  the  virtual  dis- 
placement mullipliod  by  the  sum  of  tiic  projci  (ions  on  the 
displacement  of  the  sides  of  the  polygon  Avhich  rt'pre.^ent 
the  forces  (Art  101,  Cor.).  But  the  sum  of  these  projec- 
tions is  cyual  to  thv  projection  of  the  remaining  side  of  the 
polygon,*  and  this  side  represents  the  resultant,  (Ar!.  3'6, 
Cor.  J).  Therefore,  the  sum  of  the  virtval  moments  (if  any 
number  of  concurring  forces  is  equal  to  the  virtual  moment 
of  the  resultant. 

(3)  If  the  forces  are  in  equilibrium,  their  resultant  is 
equal  to  zero  ;  hencA ,  it  follows  that  ivhen  any  number  of 
concurring  fnrren  ur»  in  equilibrium,  the  sum  of  their 
virtual  moments  =■  0. 

This  principle  la  gon(*rully  lin<»wn  as  the  Principle  of 
Virtual  Velocities,  and  Ih  of  great  use  in  the  BoluUon  of 
practical  problems  in  Statics. 

*  Krom  Uis  lUture  of  projectiuiia  (Anal.  Ocom.,  Art.  laB),  It  |8  Oleir  llilt  in  any 
terleit  of  polnU  the  projection  (on  a  glTen  line)  of  tlio  lliio  n  lili.li  Jlijna  ilio  flret  and 
l8Bt,  Ir  eq;ukl  to  the  oum  of  the  projection)!  of  the  Ilium  which  Join  the  pointH.  iwa 
and  two.  Thim,  If  the  eldes  of  a  cloxed  polfKun.  taken  In  order,  be  morkcd  witli 
arrows  pointing  from  each  vertns  to  the  next  one ;  and  If  their  projettlon-j  be 
marked  with  arrows  In  the  same  dln^ctlong,  then,  lined  mcasorsd  from  left  to  right 
being  oonsldered  pocltive,  and  linos  from  right  to  left  negative,  the  swn  (ff  tKr  pro- 
Jtetionn  <if  the  Hdet  nf  a  dvMd  pohfgon  on  any  rl(f/ii  Rm  U  atro. 
8 


n) 


nt  forces  we  may 
»f  them  first,  and 
kiit  as  above,  then 
ant  of  these  two 
'  the  rest! Hunt  of 

the  last ;  or  we 
The  sum  of  the 

the  virtual  dis- 
ojoc/iujis  on  the 
Avhich  represent 

of  those  projec- 
ining  side  of  the 
iUant,  (Art.  33, 
moments  (if  any 

virtual  moment 


eir  resultant  \» 

any  number  of 

sum  of  (heir 


tl 


Principle  of 
10  solution  of 


I-  iHf.v  tint  in  aiiy 

jiilnt  Die  flrst  and 

jiilu  the  potnti),  two 

be  marked  with 

tlii'lr  projeitlons  bo 

rsd  IW>m  left  to  right 

the  sum  of  the  pro- 


VntTDAL  MOMMJfTS. 


169 


103.  Natnre  of  the  Sisplaeemttnt.— It  must  be  care- 
fully observed  that  the  displacement  of  the  particle  on 
which  the  forces  act  ia  virtual  and  arlilrarv.  The  word 
virtual  in  Statics  is  used  to  intimate  tha^.  tbo  displacements 
are  not  really  made,  but  only  supposed,  i.  e.,  they  are  not 
actual  but  imagined  displacements  ;  but  in  the  motion  of  a 
particle  treated  of  in  Kinetics,  the  dJsplacemcni  is  often 
taken  to  be  that  which  the  particle  actually  undergoes. 
In  Art.  101,  the  displacement  was  limited  to  an  infiniteai- 
mal.  In  some  cases,  however,  &  finite  displacement  may  be 
used,  and  it  may  be  even  more  convenient  to  consider  a 
finite  displacement.  But  in  very  many  cases  any  finite  dis- 
placement is  suflScient  to  alter  the  amount  or  direction  of 
the  forces,  so  as  to  prevent  the  principle  of  virtual  velocities 
from  \mug  applicable.  This  diflSculty  can  always  bo  avoid- 
ed in  practice  by  assuming  the  displacement  to  be  infinitesi- 
mal ;  a»d  if  the  virtual  disphujement  is  infinitesimal  the 
virtual  volocitieci  are  all  inflnitcsimaL 

104    Equation  of  Virtual  Moments. —Let  P^,  P,, 

Pf,  <         !.  iiofr  the  fonos,  and  ip^.  (5jt>,,  '5/;,,  etc.,  the  vir- 
taai  I  lien  the  priut>ipUs  of  virtual  velocities  is 

expressed  (Art  102)  by  the  etiuation 

^i  •  ^Pi  +  Pt  •  ^P%  +  i\  '  ^P,  +  etc.  =  0; 

or  ^P6p  =  0,  (1) 

which  is  called  tbo  equation  of  virtu/il  moments.* 

Hon. — If  the  virtual  diHplacemont  is  at  right  angles  to 
the  direction  of  any  force,  it  is  clear  that  dp,  the  virtual 
velocity,  is  rijual  In  m-ro.  Hence,  when  the  virtual  dia- 
placement  is  at  ripht  angles,  to  the  direction  of  the  force, 

•  Or  virtual  wort  (Set  Art.  101,  Rem.).  Tbl*  eqiutlon  IiM  been  in&de  hy  La. 
itranKe  tho  fiiaudation  of  hia  Kreat  work  on  Xeclivilo,  "  MiJbaahtno  /jialytlque,'* 
(I'rlce'H  AiuO.  Uocb.,  ~.'ol.  I,  p.  14t.) 


170 


SrSTBM  or  PAItTICLXa. 


m. 


m 


»M 


the  virtual  moment  of  the  force  =  0,  and  the  force  will  not 
enter  into  the  equation  of  virtual  moments. 

Such  a  virtual  displacement  is  always  a  convenient  one 
to  choose  when  we  wish  to  get  rid  of  some  unknown  force 
which  acts  upon  a  pari:icle  or  system. 

105.  System  of  Particlea  Rigidly  Connected.— (1) 

If  a  particle  in  equilibrium,  under  the  action  of  any  forces, 
be  constrained  to  maintain  a  fixed  distance  from  a  given 
fixed  point,  the  force  due  to  the  constraint  (if  any)  is 
directed  towards  the  fixed  point 

Let  B  be  the  particle,  and  A  the  fixed  point.  Then  it  is 
clear  that  we  may  substitute  for  the  string  or  rigid  rod 
which  connecta  B  with  A,  a  smooth  circular  tube  enclosing 
the  particle,  with  the  centre  of  the  tube  at  A.  Now,  in 
order  that  B  may  be  in  equilibrium  inside  the  tube,  it  is 
necessary  that  the  resultant  of  the  forces  acting  upon  it 
should  be  normal  to  the  tul)e,  t.  e.,  direct^  towards  A. 

(2)  Let  there  be  any  number  of 
particles,  mi,  m,,  7n,,  et^.,  each 
acted  on  by  any  forces.  P,,  P,,  Pg, 
etc.,  and  connected  with  the  others 
by  inflexible  right  lines  so  that  the 
figure  of  the  system  is  invariable. 
Then  eacli  particle  is  acted  on  by  all 
the  external  forces  applied  to  it,  and 
by  all  the  internal  forces  proceeding  from  the  internal  con- 
nections of  the  particle  with  the  other  particles  of  the 
system.  Thus  tlie  particle,  m,  is  acted  on  by  Pj,  Pg,  etc., 
and  by  the  internal  forces  which  proceed  from  its  connec- 
tion with  m^,  «!,,  «t,,  etc.,  and  which  act  along  the  lines, 
mm  I,  mm^,  etc.,  by  (1)  of  this  Article.  Denote  the  forces 
along  the  lines  »iw,,  «i»i„  mm,,  etc.,  by  /j,  t^,  t^,  etc., 
and  their  virtual  velocities  by  dt^,  rf^,,  <J/,,  etc.      Now 


^  force  will  not 


convenient  one 
unknown  force 

innected.— (1) 

of  any  forces, 

from  a  given 

int  (if  any)  in 

it.  Then  it  is 
I  or  rigid  rod 
tube  enclosing 
t  A.  Now,  in 
the  tnbe,  it  is 
«ting  npon  it 
>wards  A. 


internal  oon- 
icles  of  the 
P,,  Pj,  etc., 
its  connec- 
ig  the  linos, 
e  the  forces 
/,,  <s,  etc., 
etc.      Now 


BrSTSM  OV  PART.     US8. 


171 


imagine  that  the  system  is  slightly  displaced  so  as  to 
occupy  a  new  position.  Then  (1)  of  Art.  104  gives  us 
for  n», 

I\6p^  +  P,dp,  +  etc.  +  ti&l,  +  ttStt  +  etc.    ^0,   (1) 

for  nil, 

t'JPi.  +  Pi^Pt,  +  etc.  +  /i<J<i  +  <,<»<,  +  etc  s=  0,  (a; 

proceeding  in  this  way  as  many  equations  may  be  formed  as 
there  are  particles  in  the  system. 

Now  it  is  clear  that  /,d/,,  and  /,«J^j,  in  (1)  have  contrary 
signs  from  what  they  have  in  (2).  Thus  if  the  system  is 
moved  to  the  right  in  its  displacement,  t^Sti,  and  t^it^  will 
Ix  positive  in  (1)  and  negative  in  (2)  (Art.  101),  and  hence, 
if  we  add  (1)  and  (2)  together,  thoije  terms  will  disappear ; 
in  the  same  way,  the  virtual  moment  of  the  internal  force 
along  the  line  connecting  m  with  any  other  particle  disap- 
pears by  addition,  and  the  same  is  true  for  the  interii^^l 
force  between  any  two  particles  of  the  system.  Henco, 
adding  together  all  the  equations,  the  internal  forces 
disappear,  and  the  resulting  equation  for  the  whole 
system  is 

SPdp  =  0,  (1) 

A 
■> 

and  the  same  result  is  evidently  true  whatever  be  the  num- 
ber of  particles  forming  the  system.  Hence,  if  any  num- 
ber of  forces  in  a  system  are  in  equilibrium,  the  sum  of 
their  virtual  moments  =  0. 

The  converse  is  evidently  true,  that  if  the  sum  of  the 
virtual  moments  of  the  forces  vanishes  for  every  virtual 
displacement,  the  system  is  in  equilibrium. 

The  following  are  examples  which  are  solved  by  the 
principle  of  virtual  velocities. 


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MXAMPLXS, 


EXAMPLES. 


1.  Determine  thv>  condition  of  equilibrium  of  a  heavy 
body  resting  on  a  smooth  inoline^l  plane  under  the  action 
of  given  forces. 

Let  W  bo  the  weight  of  the  body 
sustained  on  the  plane  BC  by  the 
force,  P,  making  an  angle,  0,  with  the 
plane.  To  avoid  bringing  the  un- 
known reaction,  R,  into  our  equation, 
we  make  the  displacemeLt  of  it«  point 
of  application  perpendicular  to  its  line  of  action,  (Art.  104, 
Sch.);  hence  we  conceive  0  as  receiving  a  virtual  dis- 
placement, OA,  at  right  angles  to  R,  the  magnitude  of 
which  in  the  present  case  is  unlimited.  Draw  Am  and 
A»  perpendicular  to  W  and  P  respectively,  Om  and  0» 
are  the  virtual  velocities  of  W  and  P,  (Art  101)  ;  and 
W  •  Om  and  P  •  On  are  their  virtual  moments.  Horce  (1) 
of  Art.  104,  gives 

W  •  Of»  -  -  P .  On  =  0. 


* 

^H.           But 

Om  =  OA  sia  a, 

^^H 

On  =  OA  cos  & ; 

^^BE             therefore 

W  sin  a  —  Poos*  s=  0; 

m 


which  agrees  with  Ex.  3,  Art.  41. 


:h 


If  the  Toroo  acts  parallel  to  the  plane,  9  =  0,  aud  (1) 
becomes 

P  =  W  sin  « ; 

which  agrees  with  Ex.  1,  Art.  41. 


mmemirwai 


ibrinm  of  a  heavy 
ander  the  action 


action,  (Art.  104, 
Dg  a  virtual  dis- 
thfl  magnitude  of 
Draw  Aw  and 
vely,  Om  and  0» 

(Art.  101)  ;  and 
ents.    Horce  (1) 


(1) 


«  =  0,  aud  (1) 


MXAMPLga. 


178 


2.  Sappoae  the  plane  in  Ex.  1  to  be  rough,  and  iha«  the 
body  is  ou  the  point  of  being  draggsd  up  the  pUine,  find 
the  condition  of  equilibrium. 

The  normal  resistance  will  now  be 
replaced  by  the  total  resistance,  C, 
inclined  to  the  normal  at  an  angle 
=  <f>,  tht-  argle  of  friction  (Art.  95, 
Cor. ).  Let  the  virtual  d  itplacemeut, 
OA,  take  pL>i0e  perpendicularly  to 
R,  then  (1)  of  Art.  104,  gives 

WOm  — P.On  =  0. 

But  Om  =  OA  sin  (o  -f  ^), 

i>nd  On  =  OA  cos  (^  —  6) ; 

therefore        W  aic  (a  +  0)  =:  P  oos  (^  —  9) ; 

which  agrees  with  (3)  of  Art.  96. 

?,.  Determine  the  horisontal  force 
which  will  keep  a  particle  in  a  given 
position  inside  a  circular  tube,  (1) 
when  the  bibe  is  smooth  end  (3) 
when  it  is  rough. 

(1)  Lot  the  virtual  displaoament, 
OA,  be  an  infinitesimal,  =  tis,  aloiig 
the  tube.    Then  since  dt  is  infinites* 
imal  the  virtual  velocity  of  B  =  0.    Then  the  equation  of 
virtual  moments  ij 

—  W  •  Om  4-  P  •  On  =  0. 


Wi§,» 


But 

Om  =  da  ■  sin  6, 

and 

On  =z  Ha -co*  8] 

therefore 

W«in«  =  PooiS; 

or 

P  =  W  An  e. 

EXAMPLSa. 

{%)  Snppoao  tho  force,  P,  jnst  sustains  the  particle ;  the 
normal  rcsiatanoe  must  now  be  replaced  by  the  total  resist- 
ance, making  the  angle,  ^,  with  the  normal  at  the  right  of 
it.  Take  the  virtual  displacement,  OA',  at  right  angles  to 
the  total  resistance  (Art  105,  Sch.),  and  let  it  be  as  before, 
an  infinitosimal  d*.    Then  (1)  of  Art  104,  gives 

--W-Om  +  P.0«'=0. 

But  Om  =  <fo  •  sin  (0  —  <»), 

aud  On'  =  d» '  QOi  (0  —  ^), 

therefcrs     W  •  sin  (*  —  0)  =  P  •  cos  (0  —  ^)  j 

or  P  =  W .  t»n  (fl  —  ^). 

Similarly,  if  the  force,  P,  will  just  drag  the  particle  up 
toe  tube  we  obtain 

P  =  W  •  tan  (©  +  <•). 

4.  Solve  by  virtual  velocities  Ex.  6,  Art.  62. 

Let  the  displacement  be  made  by  diminishing  the  angle 
«,  which  the  beam  makes  with  the  horiaontal  plane,  by  da, 
the  ends  of  the  beam  still  remaining  in  contact  with  the 
horizontal  end  vertical  planes.    Then  the  virtaal  velocity  of 


T  s  d •  2a  cos  a  =  ~  2a  sin  a  dm', 


aud  that  of 


W  =  (2aBinn  =  aco8a  da, 


and  those  of  the  leaotinns,  R  aUd  R',  yaoish.    Then  the 
equation  of  viitnal  momenta  if 


the  particle;  the 
by  the  total  resist- 
aal  at  the  right  of 
at  right  angles  to 
let  it  be  as  before, 
.gives 

0. 


-f); 


1  the  particle  up 


62. 

liehing  the  angle 
al  plane,  by  da, 
ontact  with  the 
irtaal  velocity  of 

dm; 


isb.    Then  th« 


MXAMPMCa.  179 

—  T2asina(;«.fWaco8«c^=rO; 
.*.    2T  sin  a  =r  W  cos  a. 

6.  Solve  Ex.  8,  Art  63,  by  virtual  velocities. 

Let  the  displacement  be  made  by  increasing  the  angle 
d  by  d9,  the  point,  A,  remaining  in  contact  with  the  wall ; 
the  virtutil  displacement  of  B  is  at  right  angles  to  the 
direction  of  the  tension,  T,  and  h'ince  the  virtual  moment 
of  T  is  zero  ;  the  virtual  velocity  of  W  is 

d  (5  cua  0  --  a  cos  0)  —  amxO  dO  —  b  wa^  d^. 
Then  (1)  of  Art  104,  gives 

yi  {a  «in  e  de  —  b  An  ift  d«l>)  —  Q\ 
.'.    &  siu  ^  (^^  =  a  sin  9  dd. 
But  from  the  geometry  vt  the  figuro  we  have 
d  sin  ^  =  3a  sin  9; 
.'.    6  cos  ^  rfi^  =  2a  COS  9  <fO; 
.-.    2tan^  =  tant; 

which,  combined  with  (6)  of  Ex.  8,  Art.  63,  gives  us  the 
values  of  sin  0  and  cos  ^ ;  and  these  in  (6)  of  that  Ex. 
give  us  the  value  of  x. 

6.  Solve  Ex.  38,  Art.  65,  by  virtual  velocities. 

■  Slow  the  bar  Is  to  rest  in  all  positioiui  on  the  curve  and  the 
peg,  its  centre  of  gnrity  will  neither  rise  nor  fall  when  the 
bar  reoeivee  a  displaouuent,  hence  Its  virtual   Telocity  hIU  s  0 ; 

•    t     MA* 


176 


EXAMPLEB. 


7.  In  Ex.  4,  Art  42,  prove  that  (1)  is  the  equation  of 
virtual  moments. 

8.  Find  the  inclination  of  the  beam  to  the  vertical  in 
Ex.  31,  Art.  66,  by  virtual  velocities. 

9.  Deduce,  by  virtual  velocities,  (1)  the  formula  for  the 
triangle  of  forces  (see  1  of  Art  32),  and  (2)  the  formula  for 
the  parallelogr&m  of  forces  (See  1  of  Art.  30). 


the  equation  of 
D  the  vertical  in 


fonnola  for  the 
I  the  formula  for 
0). 


CHAPTER    VII. 

MACHINES. 

106.  Fnnetioiu  of  a  Machine.— Ji  maehiue,  Statically, 
is  any  instrument  by  means  of  which  we  may  change  the 
direction,  magnitude,  and  point  of  application  of  a  given 
force  ;  and  Kinetically,  it  is  any  instrument  by  means  tf 
which  we  may  change  the  direction  and  velocity  of  a  given 
motion. 

In  applying  the  principle  of  virtual  velocities  to  a  system 
of  connected  bodies,  odvantige  is  gained  by  choosing  the 
virtual  displacements  in  certain  directions  (Art.  104,  Sch). 
When  we  use  this  principle  in  the  discussion  of  machines 
the  displacements  which  we  shall  choose  will  be  those  which 
the  different  parts  of  a  machine  actually  undergo  when  it 
is  etni)loyed  in  doing  worilc,  and  instead  of  equations  of 
virtual  work  we  shall  have  equations  of  actual  work;  and 
in  future  the  principle  of  virtual  velocities  will  often  be 
referred  to  as  the  Principle  of  Work.  (See  Minchin's 
Statics,  p.  383.) 

Every  machine  is  designed  for  the  purpose  of  overcoming 
ceitain  forces  which  are  called  resistances;  and  the  forces 
which  are  applied  to  the  machines  to  produce  this  effect  are 
called  movitig  forces.  When  the  machine  is  in  motion, 
every  moving  force  displaces  its  point  of  application  in  its 
own  direction,  while  the  point  of  application  of  a  resistance 
is  displaced  in  a  direotion  opposite  to  that  of  the  resistance. 
Ilenoe,  a  moving  force  is  one  whose  elementary  work  *  i> 
positive,  and  a  resistance  is  one  whose  elementary  work  is 
negative.    The  moving  force  is,  for  convenience,  called  the 

*  See  Art.  101,  Ran. 


Is, 


178 


MECHANICAL  ADVANTAOE. 


power ;  and  becanse  the  attraction  of  gravity  is  the  most 
common  form  of  the  force  or  resistance  to  bo  overcome  it  is 
usually  called  the  weight. 

The  weight  or  radstanoe  to  be  overcome  may  be  the  enrth's  attrac- 
tion, aa  in  raising  a  weight ;  the  molecular  attractiona  between  the 
particles  of  a  body  as  in  stamping  or  catting  a  metal,  or  dividing 
wood ;  or  fristlon,  aa  in  drawing  a  heavy  body  along  a  rough  road. 
The  power  may  be  that  of  men,  or  horses,  or  the  steam  engine,  etc., 
and  may  be  jost  sufficient  to  overcome  the  resistance,  or  it  may  be  in 
excess  of  what  is  necessary,  or  it  may  be  too  small.  If  just  sufficient, 
the  machine,  if  in  motion,  will  remain  uniformly  so,  or  if  it  be  at  rest 
it  will  be  on  the  point  of  moving,  and  the  power,  weight,  and  friction 
will  be  in  equilibrium.  If  thr  power  be  in  excess,  the  machine  will 
be  set  in  motion  and  will  continue  in  accelerated  motion.  If  the 
power  be  too  small,  it  will  not  be  able  to  move  the  marJiine ;  and  if 
it  be  already  in  motion  it  will  gradually  come  to  rest. 

The  geueral  problem  with  regard  to  machines  is  to  find 
the  relation  between  the  power  and  the  weight.  Some- 
times it  is  most  convenient  that  this  relation  should  be  one 
of  equality,  t.  e.,  that  the  power  should  equal  the  weight 
Generally,  however,  it  is  most  convenient  that  the  power 
should  be  very  different  from  the  weight  Thus,  if  q  man 
bus  to  lift  a  weight  of  one  ton  hanging  by  a  rope,  it  is  clear 
that  he  cannot  do  it  unless  the  mechanical  contrivance 
provided  enable  him  to  lift  the  weight  by  exercising  a  pull 
of  very  much  less,  say  one  cwt  When  the  power  is  much 
smaller  than  the  weight,  as  it  is  in  this  case,  which  is  a 
very  common  one,  the  machine  is  said  to  work  at  a  mechan- 
ical advantage.  When,  as  in  some  other  cases,  it  is  de8irul)le 
that  the  power  should  be  greater  than  the  weight,  there  is 
said  to  be  a  mechanical  disadvantage  of  the  machine. 

107.  Mechanical  Advantage.— (1)  Let  P  and  W  he 

the  power  and  weight,  and  p  and  w  their  virtual  velocities 
respectively ;  and  let  friction  be  omitted.  Then  fh>m  the 
equation  of  virtual  work  (Art  104),  we  have 

Pp-Ww=iO,     OT    ~-y 


mum 


OE. 

gravity  is  the  most 
,0  be  overcome  it  is 

be  the  eitrth's  attrac- 
itmctions  between  the 
:  «  metal,  or  dividing 
y  along  a  rough  road, 
the  Bteam  engine,  etc., 
lance,  or  it  may  be  in 
all.  If  just  sufficient, 
f  BO,  or  if  it  be  at  rest 
3r,  weight,  and  friction 
cesB,  the  machine  will 
rated  motion.  If  the 
re  the  machine ;  and  if 
>  rest. 

machines  ia  to  find 

e  weight.    Some- 

Ition  should  be  one 

equal  the  weight 

t  that  the  power 

Thus,  if  %  man 

a  rope,  it  is  clear 

nical  contrivance 

exercising  a  pull 

he  power  is  much 

|s  case,  which  is  a 

work  at  a  mechan- 

laees,  it  is  de8irul>le 

e  weight,  there  ia 

e  machine. 

iLet  P  and  »r  be 
virtual  velocities 
Then  from  the 
^ve 

10 

y 


pir'j    ,11m 


MSCHANICAL  ADVANTAQX. 


\n 


which  shows  that  the  smaller  P  is  in  comparison  with  W, 
the  smaller  w  will  be  in  comparison  with  p.  But  the 
smaller  P  is  in  comparison  with  W,  the  greate;  is  the 
mechanical  advantage.  Hence,  the  greater  the  mechanical 
advantage  is  the  lefc3  will  be  the  virtual  velocity  of  the 
weight  in  comparison  with  that  of  the  power.  Now,  if 
motion  actually  takes  place  the  vi/tual  velocities  become 
aciual  velocities ;  and  hence  we  have  the  principle  what  is 
gained  in  power  is  lost  in  velocity. 

(3)  There  are  no  cases  in  which  the  weight  and  power 
are  the  only  forceo  to  be  considered.  In  every  movement 
of  a  machine  there  will  always  be  a  certain  amount  of  fric- 
tion ;  and  this  can  never  be  omitted  from  the  equation  of 
virtual  work.  There  are  cases,  however,  as  that  of  a  balance 
on  a  knife-edge,  where  the  frict'on  is  very  small ;  and  for 
these  the  principle,  what  is  gained  in  jwwer  is  lost  in 
velocity,  is  very  approximately  true.  Where  the  friction  is 
considerable  this  is  no  longer  the  case. 

Lot  F  and  /  bo  the  resistance  of  tiiction  and  its  virtual 
velocity,  then  the  equation  for  any  machine  will  take  the 
form 

Pp  —  Ww  —  Ff  =  0, 

which  shows  us  that  although  P  can  be  made  as  small  sta  we 
wish  by  taking  p  large  enough,  yet  the  mechanical 
advantage  of  diminishing  P  is  restricted  by  the  fact  that  / 
increases  with/>;  and  therefore  as  P  diminishes  there  is' a 
corresponding  increase  of  the  work  to  be  done  against  fric- 
tion. Hence  if  friction  be  neglected,  there  is  no  practical 
limit  to  the  ratio  of  P  to  fF ;  but  if  the  friction  be  con- 
sidered, the  advantage  of  diminishing  P  has  a  limit,  since 
if  Pp  remains  the  same,  Ww  must  decrease  as  /y  increases; 
i.  c,  the  work  done  against  friction  increases  with  the^ 
complexity  of  the  machine  ;  and  thus  puts  a  practical  limit 
to  the  mcchanicAl  advantage  which  it  is  possible  to  obtain 
by  the  use  of  machines. 


180 


aiXPLB  MACBlNSa. 


108.  Sln^te  MacMnea.— The  simple  machines,  some- 
times called  the  Mec/ianical  Powers,  are  generally  enumei- 
ated  as  six  in  number  ;  the  Lever,  the  Wheel  and  Axle,  tho 
Inclined  Plane,  the  Pulley,  the  Wedge,  and  the  Screw. 
The  Lever,  the  Inclined  Plane,  and  the  Pulley,  may  bo 
considered  as  distinct  in  principle,  while  the  others  are 
combinations  of  them. 

llie  efficiency*  of  a  machiuQ  is  the  ratio  of  the  useful 
work  it  yields  to  the  whole  amount  of  work  performed  by 
it  The  use/til  work  is  that  which  is  performed  in  over- 
coming useful  resistances,  while  losi  work  is  that  wiiich  is 
spent  in  overcoming  wasteful  resistances.  Useful  lesist- 
anccs  are  those  wliich  the  machine  is  specially  designed  to 
overcome,  while  the  overcoming  of  wasteful  resistances  is 
foreign  to  its  purpose.  Friction  and  rigidity  of  cords  are 
wasteful  resistances  while  the  weight  of  the  body  to  be 
lifted  is  the  useful  resistance. 

Let  W  be  the  work  done  by  the  moving  forces,  W^  the 
useful  and  Wi  the  lost  work  when  the  machine  is  moving 
uniformly.    Then 

If  =  r.  +  Wi, 


and  if  Jf  denote  the  efficiency  of  the  machine,  we  have 

W 

In  a  perfect  machine,  whore  there  is  no  lost  work,  the 
efficiency  is  unity ;  but  in  every  machine  some  of  tho  work 
is  lost  in  overcoming  wasteful  resistances,  so  that  the 
efficiency  is  always  less  than  unity ;  and  the  object  of  all 
improvements  in  a  machine  is  to  bring  its  efficiency  as  near 
unity  as  possible. 

The  most  noticeable  of  the  wasteful  resistances  are  fric- 
tion and  rigidity  of  cords  ;  and  of  these  we  shall  consider 

*  HonmimM  ckUad  mo^Mn. 


«* 


KnviLianitTM  OF  raw  lkvsk. 


lU 


bines,  eome- 
illy  enumer- 
<id  Axle,  tho 

the  Screw, 
ley,  Tuay  bo 

otliers  arc 

f  the  useful 
erformed  by 
ned  in  over- 
lat  which  is 
'»eful  lesist- 
dedigned  to 
eaistances  is 
of  cords  are 
body  to  be 

ces,  Wm  the 
is  moving 


re  have 


work,  the 
tho  work 
that  the 
hject  of  all 
ncy  ns  near 

68  are  fric- 
II  consider 


only  the  first  The  student  who  wants  informauon  on  the 
experimental  laws  of  the  rigidity  of  cords  is  referred  to 
Weisbach's  Mechanics,  VoL  I,  p.  363. 

109.  The  Levor.—A  lever  is  a  rigid  bar,  straight  or 
onrved,  movable  about  a  fixed  axis,  which  is  called  the 
fulcrum.  The  parts  of  tho  lever  into  which  the  fulcrum 
divides  it  are  called  the  arms  of  the  lever.  When  the  arms 
are  in  a  straight  line  it  is  called  a  straight  lever ;  in  all 
other  cases  it  is  a  bent-  lever. 

Lovers  are  divided,  for  convenience,  into  three  kinds, 
according  to  the  position  of  the  falorom.  In  the  first  kind 
the  fulcrum  is  between  the  power  and  tho  weight ;  in  tho 
second  kind  the  weight  acts  between  the  fulcrum  and  tho 
power ;  in  ihe  third  kind  the  power  acts  between  the  ful- 
crum and  the  weight.  In  the  last  kind  the  power  is  always 
greater  than  the  weight 

A  pair  of  scissors  fumislies  an  example  of  a  pair  of  levers 
of  the  first  kind ;  a  pair  of  nut-crackers  of  the  second  kind ; 
and  a  pair  of  shears  of  the  third  kind. 

110.  Conditions  of  Eqnilibrinm 
of  the  Lever. — (1)  Wilh<mt  FVictton. 
Let  AB  be  the  lever  and  0  its  fulcrum ; 
and  let  the  two  forces,  P  and  W,  act  in 
the  plane  of  the  paper  at  the  points,  A 
and  B,  in  the  directions,  AP  and  BW. 
From  C  draw  CD  and  OE  perpendicular  to  the  directions 
of  P  and  W.  Let  a  and  P  denote  the  angles  which  the 
directions  of  the  forces  make  with  the  lever.  Then,  taki.ig 
moments  around  C,  w^  have 

P.CD  =  rCE, 


;i 


ii 


or 


P  _  perpendicular  on  direction  of  W 
W  "  perpendicular  on  dirootiou  of  P' 


(1) 


^^_.v  k:_     (^:j 


I 


I 


^i 


i 


188 


EqUILIBRIUU  OF  TBS  LSVJSS. 


That  is,  tho  condition  of  equilibriam  requires  that  the 
power  and  weight  should  he  to  each  otlter  inversely  as  the 
length  of  their  respective  arms  (Art.  46). 

To  find  the  pressure  on  tho  fulcrum,  and  its  dii'cctiou  ; 
let  the  directions  of  the  pressures,  P  and  W,  intersect  in 
F;  join  0  and  F;  then,  since  the  lover  is  in  equilibrium 
by  the  action  of  the  forces,  P  and  W,  and  the  reaction  of 
the  fulcrum,  the  resultant  of  P  and  W  must  be  equal  and 
opposite  to  that  reaction,  and  hence  must  prss  through  0 
and  be  equal  to  the  pressure  on  the  fulcrum.  Denott)  this 
resultant  by  R,  tho  angle  which  it  makes  with  the  lever  by 
0 ;  and  the  angle  AFB  by  <•> ;  then  we  have  by  (1)  of  Art.  30 


or 


It*  =  P*  +  W»  +  2PW COB  AFB ; 
IP  =  P»  +  W^  +  2PW cos  fc). 


(8) 


ic?iich  gives  the  pressure,  R,  on  thefukrum. 

To  find  its  direction  resolve  P,  W,  und  R  parallel  and 
perpendicular  to  the  lever,  and  we  have 

for  parallel  forces,  P  cos  a  —  W  cos  fi—R  cos  0  =  0; 

for  perpendicular  forces,  P  sm  a  +  W  sin  (i—R  sin  0  =  0; 

by  transposition  and  division  we  get 

P  sin  «  -f-  IF  sin  /3 


tan  e  = 


F  cos  a  —  W  cos  /3' 


(8) 


which  gives  the  direction  of  the  pressure. 


Cob. — When  the  lover  is  bent  or  curved  the  condition  of 
equilibrium  is  the  same. 

Solution  by  the  principle  of  virtual  velocities. 

Suppose  the  lever  to  be  turned  round  0  in  the  direction 
of  P  through  the  angle  dO,  into  the  position  ab;  let  p  and 


[aires  that  the 
iversely  as  the 

its  dii-oi^tioii ; 
V,  intersect  in 
Id  cquilibrinm 
he  reaction  of 

be  e<]Tia!l  and 

R88  through  0 

Denoto  this 

!i  the  lever  by 

(1)  of  Art.  30 


^; 


(2) 


!  parallel  and 

^  cos  0  =  0; 
i?  sin  ©  =  0; 

(3) 
condition  of 


he  direction 
let  p  and 


BQUILIBRIUM  OF  TBB  LJSVXR. 


"*1*pi 


188 


q  be  the  perpendiculars  CD  and  G£  respectively,  then  the 
virtual  velocity  of  P  will  be  (Art.  101), 

Aa  sin  a  —  AC-dO-aln  a  ■=.  pd&. 

Similarly,  the  virtual  velocity  of  TF  is  —  qdB. 
Hence,  by  the  equation  of  virtual  work  we  have 


P'P-dd-^  W-qdd  =  0; 
.-.    P  p=  W-q. 
which  is  the  same  as  (1). 


(*) 


(3)  With  Friction.  -In  the  above  wo  have  supposed  fric- 
tion to  be  neglected  ;  and  if  the  lever  turns  round  a  sharp 
edge,  like  the  scale  beam  of  a  balance,  the  friction  will  be 
exceedingly  small.  Levers,  however,  usually  consist  of  flat 
bars,  turning  about  rounded  pins  or  studs  which  form  the 
fulcrums,  and  between  the  lever  and  the  pin  there  will  of 
coni-se  be  friction.  To  find  the  friction  let  r  be  the  radius 
of  the  pin  round  which  the  lever  turns ;  then  the  friction 
on  the  pin,  acting  tangentially  to  the  surface  of  the  pin 
and  opposing  motion,  =  ^  sin  ^  (Art.  99) ;  and  the  virtual 
velocity  of  the  point  of  application  of  the  friction  =  rdO ; 
and  hence  the  virtual  work  of  the  friction  =  £  sin  ^-  rdO, 
Hence  the  equation  of  virtual  work  is 

p.pd6  —  W-qM  —  ^  sin  ^  r<W  =  0. 

Substituting  the  value  of  R  from  (2),  and  omitting  d6,  we 
havo 

Pp  —Wq  =  r  sin  fjt  ^/WT~W*  +  2Pr  cos  w ;    (5) 

solving  thie  quadratic  for  P  we  have 


X84 


TES  COMMON  BA/.AJ/CS. 


P=W 


pq  +  r'  cos  u)  giu*  ^ 


/J*  —  r*  sin*  0 


±  Wt  sin  ^ 


vV>'  4-  ^j??  COS  w  4-  g*  —  »•*  sin*  (p  sin*  ta 
p^  —  t^  siu*  flk 


.(6) 


which  gives  the  relation  between  the  power  and  the  weight 
when  friction  is  considered,  the  upper  or  lowei  sign  of 
r  sin  <)  being  taken  according  as  P  or  W  ie  about  to  pro- 
ponderate. 

Cob. — If  the  friction  is  so  small  that  it  nay  be  omitted, 
r  sin  ^  =  0,  and  (6)  becomes 


w~  p 


m 


112..  Th«  Common  Balano. — In  machines  generally 
the  object  is  to  produce  motion,  not  rest ;  in  other  words 
to  do  woik.  The  statical  investigation  shows  only  tlie  limil 
of  force  to  be  a'-plied  to  put  tiio  machine  on  the  point  of 
motion,  or  to  give  it  uniform  motion.  For  any  work  to  bo 
done,  the  force  applied  must  exceed  this  limit,  and  the 
grealor  the  excess,  the  greater  the  amount  of  worL  done. 
Thei-e  's,  however,  one  class  of  applications  of  the  levt/ 
where  the  object  is  not  to  do  work,  but  to  produce  equi- 
librium, and  which  !>,re  therefore  8))eL'iRlly  adapted  for  treat- 
ment by  statics.  This  is  the  class  of  measuring  machines, 
where  the  object  is  not  to  overcome  a  particular  resistance, 
but  to  measure  its  amount.  The  t'Osting  machine  is  a  gotnt 
example,  measuring  the  pull  which  a  bar  of  any  material 
will  sustain  Ixifore  breaking.  The  common  balance  and 
steelyard  for  weighing,  are  familiar  examples. 

The  common  balance  is  an  instrument  for  weighing ;  it 
is  a  lever  of  the  hrst  kind,  with  two  equal  anns,  with  a 
Bcale-pr.u  suspended  from  each  extremity,  the  fulcrum 
being  vertically  above  the  centrn  of  gravity  of  the  borm 
when  the  latter  is  horiscnt«l,  and  therefore  vertically  above 


lin'  0  sin*  u»  ,„. 
,  (C) 

EUid  the  weight 
lower   sign   of 
about  to  pro- 
ay  be  omitted, 


(7) 

hines  generally 
n  other  words 
3  only  the  limit 
n  the  point  of 
my  work  to  bo 
limit,  and  the 
jf  work  done. 
3  of  the  lever 
produce  equi- 
pted  for  treat- 
ing machines, 
ar  ix^sistancc, 
ine  is  a  go<Hl 
any  mittorial 
balance  and 

weighing ;  it 
arms,  with  a 

the  fulcnim 
of  the  borra 

rtically  abovo 


TBB  COMMON  BALAlWt. 


185 


the  centre  of  gravity  of  the  system  formed  by  the  beam,  the 
scale-pans,  and  the  weights  of  the  acate^paus.  The  sab- 
stunce  to  be  weighed  is  placed  in  one  ecalc-pan,  and  weights 
of  known  magnitude  are  placed  in  the  other  til)  the  beam 
remains  in  equilibrium  in  a  {)erfectly  horizontal  position, 
in  which  case  the  weight  of  the  substance  is  indicated  by 
the  weights  which  balance  it  If  these  weights  differ  by 
ever  so  little  the  horizontality  of  tbo  beam  will  be  disturbed, 
and  after  oscillating  for  a  short  time,  in  consequence  ot 
the  fulcrum  being  placed  above  the  centre  of  gravity  of  the 
system,  it  will  rest  in  a  position  inclin  -i  to  the  horizon  at 
an  a.igle,  the  extent  of  which  is  a  measure  of  the  sensibility 
of  the  balance. 

Dte  preceding  explftnatioc  raprpaenta  tlie  balanoo  in  ita  simplflet 
fom;  in  practice  there  aru  manj  modiflcationa  and  roDtrivances 
introduced.  Sfacli  akill  hso  been  expended  upon  the  cch  mrction  of 
balincea,  and  great  delicacy  haa  be.ia  olitainr'J.  Tfaoa,  the  lieam 
ehouid  l>e  auapended  hj  me&na  of  a  Icnifeedge,  i.  e. ,  a  projecting 
metallic  edge  transverse  to  its  len){th,  which  reatfl  open  a  plate  of 
agate  or  othei  hard  sulMtance.  The  chains  which  rapport  the  acaln- 
pauB  should  be  suspended  tiom  the  extremities  of  the  beam  in  the 
same  manner.  The  point  of  support  of  the  beam  (folerum)  should  be 
at  equal  distances  from  the  points  of  suspension  of  the  scales ;  and 
when  the  bnlance  is  not  loaded  the  Iteam  s.iould  be  horisontal.  We 
can  ascertain  f  these  conditions  are  satisfied  by  olwerving  whether 
tliere  is  still  'quillbrium  when  the  snbntanoe  is  transferred  to  the 
scale  which  th )  weight  originally  occupied  and  :  he  weight  to  thai 
which  the  subst  inoe  originally  oocnpind. 


The  chief  requisites  of  a  good  balance  are  : 

(1)  When  equal  weights  are  placed  in  the  sca1e-3Hui8  the 
beam  should  be  perfectly  horizontal. 

(2)  The  balance  should  poss*;?'^  great  misibilitif ;  x.  «.,  if 
two  weights  which  are  very  nearly  equal  be  placed  in  the 
Bcale-pana,  the  beam  should  vary  t$n«ibly  from  its  horizontal 
position. 


ij-i 


186 


aSQUIBITSa  OP  a  qood  balancm. 


(8)  When  the  bala..^  is  disturbed  it  should  readily 
return  to  its  state  of  rest,  or  it  should  have  staMity. 


L 


112.  To  DetenniiM  the 
Chief  RaqniBites  of  a  Good 
Balanpe. — Let  P  and  W  be 

the  weights  iu  the  scale-pans ; 
O  the  fliicrum ;  h  its  distance 
from  the  straight  line,  AB, 
which  joins  the  points  of  at- 
tachtuent  of  the  sc»le-parf>  to 

the  beam ;  O  the  centre  of  grayity  of  the  beam  ;  and  let 
AB  be  at  right  angles  to  00,  the  line  joining  the  falcmwt 
to  the  centre  of  gravity  of  the  beam.  Let  AC  =  OB  =  a; 
OG  —  k\  w  =  the  weight  of  the  beam  ;  and  6  =  tho 
angle  which  the  beam  makes  with  the  horizon  when  there 
is  equilibrium. 
Now  the  perpendicular  from  0 


on 

the  direction  of  P 

— 

a  cos 

e-h 

sin 

0', 

it 

(( 

(( 

W 

= 

a  cos 

e  +  h 

sin 

9\ 

<< 

« 

« 

to 

— 

k  sin 

e; 

therefore  tak'ng  moments  round  0  we  have 

P  (a  COS  6~h  sin  5)—  If  (a  cos  e+A  sin  6)—wh  sin  d  =  0  ; 


tan  0  = 


(P-  W)a 
{P  +  W)h  +  wh' 


a) 


This  equation  determines  the  position  of  equilibrium.  Tho 
first  requisite — tho  horizontality  of 'the  beam  when  P  and 
W  are  equtd — is  Ba^^isfied  by  making  the  arms  equal. 

The  second  requisite  [(2)  of  Art.  Ill],  requires  that,  for 
a  given  value  of  P  —  W,  the  inclination  of  the  beam  to  the 
horisoii  must  be  «s  great  as  possible,  and  therefore  the  sen- 
sibility is  greater  the  greater  tan  0  is  for  a  given  value  of 
P  —  W'y  and  for  a  given  value  of  tan  9  the  sensibility  is 


-  fietwassMsw-v-jMiMrm 


mam*mf^ffiiHKi!m^ii>slsm9S»  i'  9  ^wiv*  l 


^■IIWi'fcl|Wfl<WW*^j|g_ 


rCA 


RBQUJSITS8  Of  A   GOOD  BALANCB. 


187 


should  readily 
\tability. 


beam ;  and  let 
ig  the  fulgrnui 
C  =  CB  =  rt; 
and  B  —  tho 
pn  when  there 


sin  9; 
gin  9; 

tPit  ain  0  =  0  ; 

(1) 

ibrinm.  The 
when  P  and 
equal. 

lires  that,  for 
e  beam  to  the 
Bfore  the  son- 
iven  value  of 
sensibility  is 


greater  the  smaller  the  valae  of  P  —  W  'a\  hence  the  sen- 
sibility may  be  measured  by  -5- — ^,  which  requires  that 

be  as  small  as  possible.  Therefore  a  must  be  brge,  and  to, 
h,  and  k  must  be  small ;  t.  «.,  the  arms  must  be  long,  tuo 
beam  light,  and  the  distances  of  the  fulcrum  from  the 
beam  and  from  the  centre  of  gravity  of  the  beam  must  be 
small. 

The  third  requisite,  its  stability,  is  greater  the  greater 
the  moment  of  the  forces  which  t«nd  to  restore  the  beam  to 
iti  former  position  of  rest  when  it  is  disturbed.    1(  P=W 

this  moment  is 

[{P  +W)h  +  wk]  sin  6, 

which  should  be  made  aa  large  as  possible  to  secnre  the 
third  requisite. 

This  'M)ndition  is,  to  som<>  extent,  at  variance  with  the 
second  requisite.  Tlicy  may  both  b  .•  satisfied,  however,  by 
making  {P  +  W)h  -\-  wk  large  and  a  large  also  ;  t.  e.,  by 
increasing  the  distances  of  the  fulcrum  from  the  beam  and 
from  the  centre  of  gravity  of  the  beam,  and  by  lengthening 
the  arms.  (See  Todhunter's  Statics,  p.  180,  also  Pratt's 
Mechanics,  p.  78.) 

The  comparative  importance  of  these  qualities  of  sensi- 
bility  and  ntability  in  a  balance  will  depend  upon  the  use 
for  which  it  is  intended ;  for  weighing  heavy  woightA, 
stabilitif  is  of  more  importance;  for  use  in  a  chemical 
laboratory  the  balance  must  possess  great  BMmbiUty ;  ar  d 
instruments  have  been  constructed  which  indicate  a  varia- 
tion of  weight  less  than  a  miUionth  part  of  the  whole.  In 
a  balance  of  great  delicacy  the  fulcrum  is  made  as  thin  as 
possible ;  it  is  generally  a  kni/t-tdge  of  hardened  steel  or 
agate,  resting  on  «  polished  agate  plate,  which  is  supported 
on  a  strong  vertical  pillar  of  brass, 


t^:m^'  -1^  jvn 


188 


TBS  STEBLTASD. 


113.  The  Ste«l]rarcL— This  if  a  kiud  of  balance  in 
which  the  arras  &ve  unequal  in  length,  the  longer  one  beiuti 
graduated,  along  which  a  poist  may  be  moved  in  order  to 
balance  different  weignts  which  01*6  placed  in  a  scale-pan  on 
the  short-arm.  While  the  moment  of  the  substance 
weighed  is  changed  by  increasing  or  diminishing  its  quan- 
tity, its  arm  remaining  constant,  that  of  the  poise  is 
changed  by  altering  its  arm,  the  weight  of  the  poise 
remaming  the  same. 

114.  To  Qradnate  the  Common  Steelyard.— (1) 

When  the  point  of  suapeiunon  is  coincident  with  the  centre 
of  gravity. 

Let  AF  be  the  beam  of  the  steel- 
yard suspended  about  an  axis  pass- 
ing through  its  centre  of  gravity, 
C  ;  on  the  arm,  CF,  place  a  mov- 
able weight,  P ;  then  if  a  weight, 
W,  equal  to  P,  is  suspended  from 
A,  the  beam  will  balance  when  P 
on  the  long  arm  is  at  a  distance 

from  C  equal  to  AO.  If  W  equals  twice  the  weight  of  P, 
the  beam  will  balance  when  the  distance  of  P  from  0  is 
twice  AO ;  and  so  on  in  any  proportion.  Hence  if  W  is 
successively  1  lb.,  2  lbs.,  3  lbs.,  etc.,  the  distances  of  the 
notches,  1,  3,  3,  4,  etc.,  where  P  is  placed,  are  as  1,  2,  3, 
etc..  I.  0.,  the  arm  CF  is  divided  into  equal  divisions,  begin- 
ning at  the  fulcrum,  0,  as  the  zero  point 

(2)  When  the  point  of  suspension  is  not  coincident  toith 
the  centre  of  gravity. 

Let  C  b<»  the  fulcrum,  W  the  substance  to  be  weighed, 
hanging;  at  the  extremity.  A,  and  P  the  movable  weight. 
Suppuse  that  when  W  is  removed,  the  weight,  P,  placed  at 
B  will  balance  the  long  arm,  CF,  and  keep  the  eleelyard  in 
a  horizontal  position ;  then  the  moment  of  the  instrument 


WWiM*wr>n»-<|«ttgw>ya 


of  balance  in 
•nger  one  beiui-- 
ved  in  order  to 
1  a  scale-pan  on 
the  substance 
hing  its  quon- 
f  the  poise  is 
t  of  the  poise 


teelyard.— (1) 

vith  the  centre 


p-r^ 


n|.M 


weight  of  P, 
P  from  0  is 
lence  if  IF  is 
lances  of  the 
re  as  1,  2,  3, 
risions,  begin- 

nncident  mth 

be  weighed, 

table  weight. 

P,  placed  at 

aleelyard  iu 

instrument 


MXAMPLXa. 


189 


itself,  aboBt  0,  is  on  the  side,  CF,  and  is  equal  to  P'  CB. 
Hence,  if  W  hangs  from  A,  uid  P  from  any  point  E,  then 
for  equilibrium  we  must  have 

PCE  +  PBC  =  r.  AC;    • 

or  P-BB=fr-AO; 


BE  =  -^ .  AO. 


If  we  make  W  successively  equal  to  P,2P,  SP,  etc.,  then 
the  values  of  BE  will  be  AG,  2AC,  SAC,  etc.,  and  these 
distances  must  be  measured  off,  commencing  at  B  for  the 
zero  point,  and  the  points  so  determined  marked  1,  2,  3,  4, 
etc.  Such  a  steelyard  catinot  weigh  below  a  certain  limit, 
corresponding  to  the  first  notch,  1. 

To  find  the  length  of  the  divisions  on  the  beam,  divide 
BE,  the  distance  of  the  poise  from  the  zero  point,  by  the 
weight,  W,  which  P  balances  when  at  the  point  E.  The 
steelyard  often  has  tivo  fulcrums,  one  for  small  and  the 
other  for  large  weights. 

EXAMPLKS. 

1.  What  force  must  be  applied  at  one  end  of  a  lever 
12  ins.  long  to  raise  a  weight  of  30  lbs.  hanging  4  ins.  from 
the  fulcrum  which  is  at  the  other  end,  and  what  is  the 
pressure  on  the  fulcrum  f  Ans.  10  lbs. :  20  lbs. 

2.  A  lever  weighs  3  lbs.,  and  its  weight  acts  at  its  middle 
point ;  the  ratio  of  its  arras  is  1 :  3.     If  a  weight  of  48  lbs. , 
be  hung  from  the  end  of  the  shorter  arm,  what  weight 
must  be  suspendeu  from  the  other  end  to  prevent  motion  ? 

Ans.  15  lbs. 


f 


HI 


IW 


WHEEL  Am)  AXLM. 


8.  The  arms  of  a  bent  lever  are  3  ft.  and  S  ft.  and  inclined 
to  each  other  at  an  angle  B  =  150°.  To  the  short  arm  a 
weight  of  7  lbs.  is  applied  and  to  the  long  osm  a  weight  of 
6  lbs.  is  applied.  Required  the  inclination  of  each  arm  to 
the  horizon  when  there  is  equilibrium. 

Ans.  The  short  arm  is  inclined  at  an  angle  of  18°  32' 
above  the  horizon,  and  the  long  arm  is  inclined  at  an  angle 
of  48°  22'  below  the  horizon. 


1.15.  The  Wheel  and  Axle.— This 
machine  consists  of  a  wheel,  a,  rigidly 
connected  with  a  horizontal  cylinder, 
b,  movable  round  two  tmnnions  (Art 
99),  one  of  which  is  shown  at  c.  The 
power,  P,  is  applied  at  the  circumfer- 
ence of  the  wheel,  sometimes  by  a  cord 
coiled  round  the  wheel,  sometimes  by 
handspikes  as  in  the  capstan,  or  by 
handles  as  in  the  windlass ;  the  weight,  W,  hangs  at  the 
end  of  a  cord  Listened  to  the  axle  and  coiled  round  it 


116.  Conditions  of  EqcUibrinm  of  tiie  Wheel  and 
Axle. — (1)  Let  a  and  b  be  the  radii  of  the  wheel  and  axle 
respectively  ;  P  and  W  the  power  and  weight,  supposed  to 
act  by  strings  at  the  circumference  of  the  wheel  and  axle 
perpendicular  to  the  radii  a  and  b.  Then  either  by  the 
principle  of  virtual  velooiti  <  or  by  the  principle  of  momenta 
we  have 

Pa    =     Wby 


or 


P  _   radius  of  axle 
W  ~  radius  of  wheel' 


(1) 


It  is  evident  that,  by  increasing  the  radius  of  the  wheel 
or  by  diminishing  the  radius  of  the  axle,  any  amount  of 
mechanical  advantage  may  be  gained.    It  will  also  be  seen 


5  ft.  and  inclined 

the  short  arm  a 

ium  a  weight  of 

D  of  each  arm  to 

angle  of  18°  22' 
lined  at  an  angle 


V,  hang8  at  the 
round  it 

he  Wheal  ftud 

wheel  and  axlo 
;ht,  sapposed  to 

wheel  and  axle 
m  either  by  the 

pie  of  moments 


(1) 

ioB  of  the  wheel 
any  amount  of 
rill  also  be  seen 


DTFTERSIfTIAL   WBJBSL  AlfD  AXLB. 


191 


that  thi"  machine  is  only  a  modification  of  the  lever ;  the 
peculiar  advantage  of  the  wheel  and  axle  being  that  an  end< 
lc'88  scries  of  levers  are  brought  into  play.  In  this  respect, 
then,  it  surpasses  the  common  lever  in  mechanical  advan- 
tage. 

In  the  above  we  have  supposed  friction  to  be  neglected, 
or,  what  amoui.f«  to  the  same  thing,  have  assumed  that  the 
trunnion  is  indefinitely  small  In  practice,  of  course,  the 
trunnion  has  a  certain  radius,  r,  and  a  certain  coefficient  of 
friction.  Calling  R  the  resultjint  of  P  and  If,  and  taking 
into  account  the  friction  on  the  trunnion  we  have  for  the 
relation  between  P  and  W  • 

Pa  =  ITJ  +  r  sin  ^  V/^lPrMP  1Fcm"w,     (2) 

u  being  the  angle  between  the  directions  of  P  and  W 
exactly  as  in  Art  110. 

(2)  Differential  Wheel  and 
Axle. — By  diminishing  b,  the  radius 
of  the  axle,  the  strength  of  the 
machine  is  diminished ;  to  avoid  this 
disadvantage  a  differential  wheel  and 
axle  is  sometimes  employed.  In  this 
instrument  the  axle  consists  of  two 
cylinders  of  radii  h  and  b' ;  the  rope 
is  wound  round  the  former  in  one 
direction,  and  after  passing  under  a 
movable  pulley  to  which  the  weight 
is  attached,  is  wound  round  the  latter  in  the  opposite  direc- 
tion, so  that  as  the  power,  P,  which  is  applied  ae  before, 
tangentially  to  the  wheel  of  radius,  a,  moves  in  its  own 
direction,  the  rope  at  h  winds  up  while  the  rope  at  b'. 
unwinds. 

For  the  equilibrium  of  the  forces  (whether  at  rest  or  in 
uniform  motion),  the  tensions  of  the  rope  in  hn  and  b'n 


Pia.eo 


■:>vaiisimmtmg^ 


192 


TOOTHED  WBSBLa. 


are  each  equal  \Ai\W.    Hence,  taking  moments  ronod  the 
centre  of  the  traniiion,  e,  we  have 


Pa  ■{- \Wh'  -  \m  ^  Oi 


(3) 


hence  by  making  the  difference,  b  —  b',  small,  the  power 
can  be  made  as  small  as  we  please  to  lift  a  given  weight. 
Let  the  wheel  turn  through  the  angle  dd;  the  point  of 
application  of  P  will  describe  a  space  :=  add,  and  the 
weight  will  bo  liftad  through  a  space  =  ^  (6  —  b')  de, 
which  latter  will  be  very  small  if  b  —  b'  is  very  small. 
Therefore,  since  the  amount  of  taork  to  be  done  to  raise  the 
weight  to  any  given  height,  is  constant,  economy  of  power 
is  accomplished  by  a  loss  in  the  time  of  performing  the 
work. 

117.  Toothed  Whoels. — Toothed  or  cogged  toheeh  arc 
wheels  provided  on  the  circumferences  with  projections 
called  teeth  or  cogs  which  interlock,  as  shown  in  the  figure, 
and  which  are  therefore  capable  of  transmitting  force,  so 
that  if  one  of  the  wheels  be  turned  round  by  any  means, 
the  other  will  be  turned  round  also. 

When  the  teeth  are  on  the  aides  of  the  wheel  instead  of 
the  circumference,  they  are  oalldd  croton  wheels.  Wlicn 
the  axes  of  two  wheels  are 
neither  perpendicular  nor 
parallel  to  each  other,  the 
wheels  take  the  form  of 
frustums  of  cones,  and  are 
called  beveled  wheels.  When 
tliero  is  a  pair  of  toothed 
wheels  on  each  axle  with  the 
teeth  of  the  large  one  on  one 
axle  fitting  between  the  teeth  .       ifq    Fi».M 


InaiA 


^^^^MmMMm 


j.i;j«a:n«tPn*r.rf-i' 


.:^^^R~^^l^-^ 


nomeDts  roaod  the 


0; 


(3) 


,  small,  the  power 
ifb  a  given  weiglit. 
<W;  the  point  of 
3  ::=  add,  and  the 
9  ==  ^  (A  _  b)  60, 
-  V  is  very  small, 
e  done  to  raise  the 
Bconomy  of  power 
>f  performing  the 

cogged  wheels  arc 
with  projectiona 
awn  in  the  figure, 
smitting  force,  so 
id  by  any  means, 

wheel  instead  of 
wheels.     When 


f\%.U 


TOOTHED   WHEELS, 


193 


of  the  small  one  on  the  next  axle,  the  larger  wheel  of  each 
pair  is  called  the  wheel,  and  e  smaller  is  called  the  pinion. 
By  means  of  a  combination  of  toothed  wheels  of  this  kind 
called  a  train  of  wheels,  motion  may  be  transferred  from 
one  point  to  another  and  work  done,  each  wheel  driving 
the  next  one  in  the  aeries.  The  discussion  of  this  kind  oic 
machinery  possesses  great  geometric  elegance  ;  but  it  would 
be  out  of  place  in  this  work.  We  shall  give  only  a  slight 
sketch  of  the  skimpiest  case,  that  in  which  the  axes  of  the 
wheels  are  all  parallel.  For  the  investigation  of  the  proper 
formfe  of  teeth  in  order  that  the  wheels  when  made  shall 
run  truly  one  upon  another  the  student  is  referred  to  other 
works,* 

118.  To  Find  the  Relation  of  the  Power  find 
Weight  in  Toothed  Wheels.— Let  j*  and  B  be  the  fixed 
centres  of  the  toothed  wheels  on  the  circumferences  of 
which  the  teeth  are  arranged ;  QCQ  a  normal  to  the  sur- 
faces of  two  teeth  at  their  point  of  contact,  C.  Suppose  an 
axle  is  fixed  on  the  wheel,  B,  and  the  weight,  W,  suspended 
from  it  at  E  by  a  cord  ;  also,  suppose  the  power,  P,  acts  at 
D  with  an  arm  DA;  di-aw  Aa  and  BA  perpendicular  to 
QCQ.  Let  Q  be  the  mutual  pressure  of  one  tooth  upon 
another  at  C ;  this  pressure  will  be  in  the  direction  of  the 
normal  QCQ.  Now  since  the  wheel,  A,  is  in  equilibrium 
about  the  fixed  axis.  A,  under  the  action  of  the  forces, 
P  and  Q,  we  have 

P.AD  =  ^.Aa;  (1) 

and  since  the  wheel,  B,  is  in  equilibrium  abont  the  fixed 
axis,  B,  under  the  action  of  the  forces,  Q  and  W,  we  have 


Fr.BE=:  ^B*. 


(2). 


•  8m  Ouodere'i  OimmU  tf  MtOmUm ;  Rankice's  AppHtd  MetAanie* ;  Mom. 
ley's  Oiglnetrbtg;  WUlls'*  PHmetplm  tf  UtehamUm;  CoUlgnon's  SloMfiM,-  and 
•  Paf$r  qf  Mr.  Mtf*  <«  M<  Om*.  mi.  Trmt.,  Vol.  n,  p.  fn. 
9 


"F 


194 


TRAIN  OP  IT    WaSBLa. 


Dividing  (1)  by  (2)  we  have 
P 


W 


AD 
BE 


Aa 
B4' 


or 


moment  of  P  _  Aa 
moment  of  Jf  ~  B4* 


If  the  direction  of  the  normal,  QCQ.  at  the  point  of  con« 
tact,  C,  changes  ae  the  action  passes  from  one  tooth  to  the 
succeeding,  the  relation  of  P  to  TT  becomes  variable.  But, 
if  the  teeth  are  of  such  form  that  the  normal  at  their  point 
of  contact  shall  always  be  tangent  to  both  wheels,  the  lines 
Aa  and  Bd  will  become  radii,  and  their  ratio  constant. 
And  since  the  number  of  teeth  in  the  two  wheels  is  propor- 
tional to  their  radii,  we  have 


moment  of  P  _  number  of  teeth  on  the  wheel  P 
moment  of  fT ""  number  of  teeth  on  the  wheel  W' 


(3) 


119.  Relation  of  Power  to  Weight  in  a  Train  of  n 
Wlieela — Let  R^,  R,,  R^,  etc.,  be  the  radii  of  the  suc- 
cessive wheels  in  such  a  train ;  Ti,  r„  r„  etc.,  the  radii  of 
the  corresponding  pinions;  and  let  P,  P,,  P,,  P,,  .  .  .  W, 
be  the  powers  applied  to  the  circumferences  of  the  successive 
wheels  and  pinions.  Then  the  first  wheel  is  in  equilibrium 
about  its  axis  under  the  action  of  the  forces  Pand  P.^, 
since  the  power  applied  to  the  circumference  of  vhe  second 
wheel  is  equal  to  the  reaction  on  the  first  pinion,  therefore 


Similarly 


p 

X  R^ 

= 

Pi 

X  r,. 

Pt 

A  R, 

= 

P, 

X  r, ; 

p. 

X  /?, 

= 

P, 

X  r, ; 

etc 

= 

etc.; 

Pn-l 

X  Rn 

=3 

w 

X  r» 

•■  .  ■  jpaiMiy^jiiJMLwiMiik^mi!^^ 


mxisenmmsswf&'i^sirx: 


m. 


EXAMPLES. 


195 


the  point  of  con- 
Q  one  tooth  to  the 
IBS  variable.  But, 
mal  at  their  point 
)  wheels,  the  lines 
'ir  ratio  constant. 
I  wheeUi  ia  propor- 

khe  wheel  P     .  . 
he  wheel  W'   ^  ' 

in  a  Train  of  n 

radii  of  the  sac- 
!.,  the  radii  of 
Pi.  Ps,  .  .  .  W, 
of  the  successive 
in  equilibrium 
jrcea  Pand  Pj, 
ce  of  the  second 
inion,  therefore 


is 


Multiplying  these  equations  together  and  omitting  common 
factors,  wo  have 


P 
W 


_    '1 


X   r,  X    r.  X 


•     ■     a     • 


Jii  X  R,  X  Jti  X 


(1) 


It  will  be  observed,  in  toothed  gearing,  that  the  smaller 
the  radius  of  the  pinion  as  compared  with  the  wheel,  the 
greater  will  be  the  mechanical  advantage.  There  is,  how- 
ever, a  practical  limit  to  the  size  that  can  be  given  to  the 
pinion,  because  the  teeth  must  be  large  enough  for  strength, 
and  must  not  be  too  few  in  number.  Six  is  generally  the 
least  number  admissible  for  the  teeth  of  a  pinion.  Equa- 
tion (1)  shows  that  by  a  train  consisting  of  a  very  few  pairs 
of  wheels  and  pinions  there  is  an  enormous  mechanical 
advantage.  Thus,  if  there  are  three  pairs,  and  the  ratio  of 
each  wheel  to  the  pinion  is  10  to  1,  then  P  is  only  one 
thousandth  part  of  W;  but  on  the  other  hand,  W  will  only 
make  one  turn  where  P  makes  one  thousand.  Such  trains 
of  wheels  are  very  useful  in  machinery  such  as  hand  cranes, 
where  it  is  not  essential  to  obtain  a  quick  motion,  and 
where  the  power  available  is  very  small  in  comparison  to 
the  weight    (See  Browne's  Mechanics,  p.  109.) 

EXAMPLES. 

1.  What  is  the  diameter  of  a  wheel  if  a  power  of  3  Ibe. 
is  just  able  to  move  a  weight  of  12  lbs.  that  hangs  from  the 
axle,  the  radius  of  the  axle  being  2  ins.?       Arts.  16  ins. 

2.  If  a  weight  of  20  lbs.  be  supported  on  a  wheel  and 
axle  by  a  force  of  4  lbs.,  and  the  radius  of  the  axle  is 
f  in.,  find  the  radius  of  the  wheeL  Ana.  3^  ins. 

3.  A  capstan  is  worked  by  a  man  pushing  at  the  end  of 
a  pole.  He  exerts  a  force  of  50  lbs.,  and  walks  10  ft. 
round  for  every  2  ft.  of  rope  pulled  .in.  What  is  the 
icsistauce  overcome ?  Ans.  250  lbs. 


196 


tNCUi^SD  PLAlfE. 


4.  An  axle  whose  diameter  is  10  ine.,  iias  on  it  two 
wheels  the  diameters  of  wliich  are  2  ft  and  2|  ft.  respew- 
tively.  Find  the  weight  that  would  be  suppoi  ted  on  the 
axle  by  weights  of  25  lbs.  and  24  lbs.  on  the  smaller  and 
lurgor  wheels  respectively.  Ana.  264  lbs. 

120.  The  Inclined  Plane.— This  has  already  been 
partly  considered  (Art  96,  etc.).  Let  the  power,  P,  whose 
direction  makes  an  angle,  B,  with  a  rough  inclined  plane, 
be  employed  to  drag  a  weight,  W,  up  the  plane.  Then  if 
0  is  the  angle  of  friction  and  i  the  inclination  of  the  plane, 
we  have  from  (3)  of  Art.  96, 


P  _  wwn  (t  4-  »)      • 
"^  cos  («  -  e)' 

If  P  acts  along  the  plane,  0  =  0,  and  (1)  becomes 

p  -  |yB'P(»'  +  ^), 

cos  ^ 
If  P  acts  horizontally,  9  =  —  t,  and  (1)  becomes 
P=  ff'  tan  («•  +  0). 


(1) 


(2) 


(8) 


Cor.— If  we  suppose  the  friction  =  0,  (1),  (3),  and  (3) 
become  respectively 

rsin  t 


P=  W 


COS0* 


<*) 


P=  TTsini,  (5) 

P  =  fF  tan  i.  (6) 

SoH.-— It  follows  from  (4),  (5),  and  (6)  that  the  smaller 


Mituitttsus»<>iititmmmm 


ms.,  lias  un  it  two 
ft  aud  2|  ft.  respac- 
*o  Buppoited  on  the 
on  the  smaller  uud 
Ann.  264  lbs. 

}  has  already  boon 
the  power,  P,  whose 
agh  inclined  plane, 
he  plane.  Then  if 
oation  of  the  plane, 


(1) 


(1)  becomes 


(2) 

(1)  becomes 

(3) 
(1),  (2),  and  (3) 

(4) 

(6) 

(6) 
that  the  smaller 


— S! 


w^mmmmm. 


TBE  PULLEY. 


mr- 


197 


the  inoKtiation*  of  the  plane  to  the  horison,  *he  greater  will 
be  the  mechanical  advanta^.    If  we  take  in  friction  there 

exception  to    this    rule    wnen   i  >  ^  —  0.      The 


18  an 


2 


gradients  on  railways  are  the  most  common  examples  of 
the  use  of  the  inclined  plane ;  these  are  always  made  au  low 
iis  is  convenient  in  order  to  enable  the  engine  to  lift  the 
lieavieat  possible  train. 

121.  The  Pnlley. — The  pulley  consists  of  a  grooved 
wheel,  capable  of  revolving  freely  abo><t  an  axis,  fixed  into 
a  framework,  called  the  block.  A  cord  passes  over  a  por- 
tion of  the  circumference  of  the  wheel  in  the  groove. 
Wlien  the  axis  of  tlie  pulley  is  fixed,  the  pulley  is  called  a 
Jixed  pulley,  and  its  only  eflTect  is  to  change  the  direction 
of  the  force  exerted  by  the  cord ;  but  where  the  pulley  can 
ascend  and  descend  it  is  called  a  movable  pulley,  and  a 
mechanical  advantage  may  bo  gained.  Combinations  of 
pulleys  may  be  made  in  endless  vaxnety;  we  shail  consider 
only  the  simple  movable  pulley  and  three  of  the  more 
ordinary  combinations.  No  account  will  be  here  taken  of 
the  weight  of  the  pulleys  or  of  the  cord,  or  of  friction  and 
stiflfness  of  cords.  The  weight  of  a  set  of  pulleys  is  gener- 
ally small  in  comparison  with  the  loads  which  they  lift ; 
aud  the  friction  is  small.  The  use  of  the  pulley  is  to 
diminish  the  effects  of  friction  which  it  does  by  transferring 
the  friction  between  the  cord  arid  circumference  of  the 
wheel  to  the  axis  and  its  supports,  which  may  be  highly 
polished  or  lubricated.  The  mechanical  principle  involved 
in  all  calculations  with  respect  to  the  pulley  is  the  constancy 
of  the  force  of  tension  in  all  parts  of  the  same  string 
(Art.  40).  

•  To  And  the  incIitiAtion  of  the  plane  for  a  maxlniiim  value  of  P  when  It  acts 
parallel  to  the  plane  we  pnt  Ihe  derivative  of  P  with  respect  to  J  =  0,  and  get 

!''*  =  IF  '^°'  ^'  *-♦]!  =  0,  .-.  I  =  "  -  *.    Hence  whilfl  the  incllnatton  of  the  plane 
dl  coa^  % 

18  dimlnisbiug  from  '  to  ^  -  ^,  mociumical  advantoie  U  diministaiug. 


198 


riRST  araTBM  of  pullets. 


12?.  The  Simple  Movable  Pulley.— Let  0  be  the 

centre  of  the  pulley  which  is  supported  by  a  cord  passing 
un  ^er  it  with  cue  end  attached  to  a  beam  at  A  and  the 
other  end  stretched  by  the  force  /. 

Now  since  the  tension  of  the  string, 
ABDP,  is  the  same  thoughont,  and  the 
weight,  W,  io  supported  by  the  two 
strings  at  B  and  D,  in  each  of  which 
the  tension  is  P,  re  have 


2P=  W', 


P 
W 


1 
2* 


The  same  result  follows  by  the  prin- 
ciple of  virtual  Yelot^ities.    Suppose  the 
pulley  and  tbe  weigbt,  W,  to  rise  any 
distance.     Then  it  is,  clear  that  both  halves  of  the  string 
must  be  shortened  by  the  same  distance,  and  hence  P 
must  rise  double  the  distance ;  and  therefore  the  equation 
of  virtual  work  gives 

P       1 

The  mechanical  advantage  with  a  single  movable  poUoy 
is  2. 


123.  First  Bjwxma  of  PnUeye,  in  which 
the  same  curd  passes  round  all  the  Fal- 
leya. — In  this  systcitn  there  arc  two  blocks,  A 
and  R,  the  upper  of  which  is  fixed  and  the 
lower  movable,  and  each  containing  a  num^^r 
of  pulleys,  each  pulle>  being  movable  round 
the  axis  of  the  block  in  v/hioh  it  is.  A  single 
cord  is  attached  to  the  lower  block  and  passes 
alternately  round  thi  pulleys  in  the  np|)er  and 
lower  hlooks,  the  portions  of  the  cord  l)otwoon 
successive  pulleys  being  iHtrallul.    The  (Kirtiou 


ijfrjs. 


»y.— Let  0  be  the 
L  by  a  cord  passing 
beam  at  A  and  the 


^ 


1 


© 

Fig.e2 

Hives  of  the  string 
nee,  and  hence  P 
efore  the  equation 


lo  raorable  pulley 


IIS8T  arsTMM  or  pullbts. 


199 


of  cord  proceeding  from  one  pulley  to  the  next  is  called  a 
ply;  the  portion  at  ivhioh  the  power,  P,  is  applied  is 
called  the  taokli-faU. 

Since  the  cord  passes  round  all  the  pulleys  its  tension  is 
the  rame  throughout  and  equal  to  P.  Then  if  n  be  the 
number  of  plies  at  the  lower  b^ock,  nP  will  be  the  resultant 
upward  tension  of  the  cords  at  the  lower  block,  which 
must  equal  W ; 

.-.    nP  =  W, 


or 


P 
W 


1 
n 


This  result  follows  also  by  the  principle  of  virtaal  yeloci- 
ties.  Let^  denote  the  length  of  the  tockle-fall  and  x  the 
common  length  of  the  plies ;  then  since  the  length  of  the 
cord  is  constant,  we  have 


/»  +  »*»: 

—  cor 

istant 

• 

•  ••  dp  + 

ndx  : 

=  0. 

But  the  equation 

of  virtual  work 

is 

Pdp  + 

Wdx 

=  0; 

• 
•     • 

--1 

or 

P 

w~ 

1 

n 

This  system  it*  most  commonly  used  on  aooonnt  of  itR 
superior  portability  and  is  the  only  one  of  practical  impor- 
tance. The  several  pulleys  are  usually  mounted  on  a  com- 
mon axis,  as  in  the  figure,  the  cord  being  inclined  slightly 
asido  to  rmss  from  one  pair  of  pulleys  to  the  next. 

This  forms  what  is  called  a  set  of  BloJcs  and  Falh.  It 
is  very  commonly  used  on  shipboard  and  wherever  weight* 
have  to  be  Hfted  at  irregular  times  and  places.  The  weight 
of  the  lower  set  of  pulleys  in  this  caae  merely  forms  part  of 
the  groM  weight  W. 


300 


SBCOND  SrSTSX  OF  PULLMTS. 


Tho  friction  on  the  spindle  of  any  particnlar  pulley  is 
proportional  to  the  total  pressure  ^n  the  pulley,  which  is 
clearly  %P.  Hence,  if  fi  is  the  coefficient  of  faction,  the 
resistance  of  iViclion  on  any  pulley  =  2P/i;  and  the 
amount  of  its  displacement,  when  W  is  raised,  will  be  to 
the  displacement  of  W  in  the  ratio  of  the  radius  of  the 
epindle  to  that  of  the  pnlley. 


124  Second  Bywltmm  of  Fnlloys, 
in  which  each  Pnlley  lumffi  from  a 
fixed  block  by  a  sepanite  SKring.— < 

Let  A  be  the  fixed  pulley,  n  the  number 
of  movable  pnileys ;  each  cord  has  one 
end  attanb<Ml  to  a  fixed  point  in  the  beam, 
and  all  except  the  lust  have  the  other  end 
attached  to  a  movable  pulley,  the  por- 


Fi9.M 


first 


tions  not  in  contact  with  any  pulley  being  all  ]>a-^lle1. 
Then  the  tension  of  the  cord  passing  under  the 

W 

(lowest)  pulley  =  -^  (Art.  128) ;  the  tension  of  the  cord 

W 

passing  under  tho  second  pulley  =  -^,  and  so  on  ;  and  the 

W 

tension  of  the  cord  passing  uiider  tho  nth  pulley  =  ^, 

which  must  equal  the  power,  P\ 

P  _  1 


0) 


i 


The  same  result  follows  by  the  principle  of  work.  Sup- 
pose tho  first  pnlley  and  the  weight  W  to  rise  any  distan*^ 
« ;  then  it  is  clear  ^hat  both  portions  of  the  oord  passing 
round  this  pulley  will  be  shortened  by  the  same  distance, 
and  hence  tlio  second  ])nIloy  must  rise  double  this  dlst^moe 
or  9-",  and  the  third  pulley  must  rise  double  the  disfjtnce  of 
the  second  or  2*x,  and  so  on  ;  and  the  nth  pulley  must  rise 
%*-''^  and  P  must  dvsvcud  "if^x ;  therefore  the  work  of  P 


Mk 


ticnlar  pulley  iu 
iallej,  which  is 
of  friction,  the 
IPft ;  and  the 
used,  will  be  to 
)  radins  jf  the 


/\^ 


under  tho  first 
on  of  the  cord 

so  on  ;  and  the 

pulley  =  ^, 


(l) 

)f  work.  3up- 
le  any  distame, 
e  oord  paasing 
dame  diatance, 
this  distimoe 
the  distance  of 
ulloy  muHt  rise 
the  work  of  P 


THIRD  arSTSM  Of  PULLKY8. 


201 


is  P2»a;,  and  the  work  to  be  done  on  fF  is  W-x.     Hence 
the  equation  of  work  gives 

P.2-«=FF.V    ...    |,=  |,. 

125.  Third  SyBtem  of  Pnlleya,  in  which  each  cord 
ia  attached  to  the  weight— In  this  aystem  oue  end  of 
each  cord  is  attached  to  the  bar  from  which  the  weight 
hangs,  and  the  other  supports  a  pulley,  the  cords  being  all 
])arallel,  and  the  number  of  movable  pulleys  one  less  than 
the  numboi  of  cords. 

Let  n  bo  the  number  of  cords;  then  the 
tension  of  the  cord  to  which  P  is  attached  is 
r ;  the  tension  ot  the  second  cord  is  2P  (Art 
122) >  t^^at  of  the  next  2>P,  and  so  on;  and 
the  tension  of  the  »th  cord  is  2»-»P.  Then 
the  sum  of  all  the  tensions  of  the  cords 
attached  to  the  weigl  t  must  equal  W. 
Hence 


/»  +  2P  +  2»P  + 


2»-»P  = 


2"  —  1 


In  thifl  system  the  weights  of  the  movable  pulleys  assist  P ; 
in  the  two  former  systems  they  act  against  it. 

(CXAMPL.ES. 

1.  What  force  ia  necessary  to  raise  a  weight  of  480  lbs. 
by  an  arrangement  of  six  pulleys  in  which  the  same  string 
passes  round  each  pulley  ?  Ana,  80  lbs. 

* 

8.  Find  the  power  which  will  support  a  weight  of 
800  lbs.  with  three  movable  pulleyai,  arranged  as  in  the 
second  system.  An*.  lOOlbs. 


inOS 


TWf  WMDOU, 


3.  If  thero  b«  eqoiUbriam  between  P  and  W  with  three 
pulleys  in  the  third  syBtem,  what  ftdditional  weight  can  be 
raised  if  2  lbs.  be  added  to  P?  Ana.  14  lbs. 

126.  The  Wedg*. — The  wedge  w  a  triangnlar  prism, 
usually  isosceles,  and  is  used  for  separating  bodies  or  parts 
of  the  same  body  by  introdacing  its  edge  between  them  and 
then  thrusting  the  wedge  forward.  This  is  effected  by  the 
blow  of  a  hammer  or  other  snoh  moaii^  which  produces  a 
▼iolent  pressure,  for  a  short  time,  in  a  direction  perpen- 
dicular to  the  back  oi  the  wedge,  and  the  resistance  to  be 
overcome  consists  of  friction  and  a  reaction  due  to  the 
moleoalar  attractions  of  the  particles  of  the  body  which 
are  being  separa>>ed.  This  reaction  will  be  in  a  direction 
perpendicular  to  the  inclined  surface  of  the  wedge. 

137.  The  lfeet4aio«l  Ad- 
vantage of  the  Wedge.-^Let 
ACB  represent  a  section  of  the 
wedge  perpendicular  to  its  in- 
clined fitces,  the  wedge  having 
been  driven  into  the  material  a 
distauce  equal  to  DC  by  a  force, 
P,  acting  in  the  direction  DO. 
Draw  DE,  DF,  perpendicular  to 
AC,  BC,  and  let  B  denote  the 
reactions  along  ED  and  FD ;  then  liR  will  he  the  frictitm 
acting  at  E  and  F  in  the  directions  EA  and  FB.  Let  the 
angle  of  the  wedge  or  ACB  =  %a. 

Resolve  the  foroos  which  act  on  the  wedge  in  directions 
perpendioalar  and  parallel  to  the  back  of  the  wedge,  then 
we  have  for  peqiendioular  forces 


n9.N 


P  =  2R  sin  a  +  2ftR  co»  a. 


(1) 


7%M  equaiioH  may  ako  i»  oHmntd  from  Uie  prinoiple  of 
work  m  follows :    If  the  wedge  has  been  driv«B  inW  the 


W  with  tbree 
weight  ean  be 
Ana.  14  lbs. 

ingubir  prism, 
odiee  or  parts 
reen  them  and 
iflected  by  the 
ch  produces  a 
action  perpen- 
sistauoe  to  be 
n  due  to  the 
ic  body  which 
in  a  direction 
redge. 


I9.M 

the  frictioB 
i'B.     Let  the 

in  directions 
wedge,  than 


(1) 
fmnoiple  tf 


MBOHANICAL  ADVANTAatt  OP  WJBDGX. 


203 


material  a  distance  eqnal  to  DO  by  a  force,  P,  acting  in 
the  direction  DO,,  then  the  work  done  by  i*  is  P  x  DO 
(Art  101,  Rem.);  and  since  the  points  £!  and  F  were 
originally  together,  the  work  done  against  the  resistance 
i?  is  22  X  DE  -f  /?  X  DP  =  a^  X  DE ;  and  the  work 
done  against  friction  is  inR  x  EC.  Hence  the  equation 
cf  work  is 

P  X  DO  =  2i2  X  DE  +  2^i2  x  EC,  (?) 

which  reduces  to  (1)  by  substituting  sin  a  and  cos  a  for 

DE      -  EC 

]T0"*'^D0* 

Cob. — If  friction  be  neglected,  (2)  becomes 


P 


2DE 
DC 


AB 
AC 


that  is        -i:  = 


P 

R 


back  of  the  wedgg 

length  of  one  of  the  equal  sides' 


It  follows  that  the  narrower  the  back  of  the  wedge,  the 
greater  will  be  the  mechanical  advantage.  Knives,  chisels, 
and  many  other  implements  are  examples  of  the  wedge. 

In  the  action  of  the  wedge  a  great  part  of  the  power  is 
employed  in  cleaving  the  material  into  which  it  is  driven. 
The  force  required  to  effec^:  this  is  so  great  that  instead  of 
applying  a  continuous  pushing  force  perpendicular  to  the 
back  of  the  wedge,  it  is  driven  by  a  series  of  blows.  Be- 
tween the  blows  there  is  a  powerful  reaction,  B,  acting  to 
puBh  the  weJge  back  again  out  of  the  cleft,  and  this  is 
resisted  uj  the  triction  which  now  acts  in  the  directions 
EC  and  FO.  tience  when  the  wedge  is  on  the  point  of 
starting  back,  between  the  blows,  t.a  equation  of  equi- 
librium will  be  from  (1) 

2S  sin  a  —  2ftR  cos  a  =r  0 ; 

.  • .     as  tAn~'  I*. 


S04 


THM  8CBBW. 


fi  N 


And  tho  wedge  will  fly  back  or  not  according  as  a  >  or 
<  tan~*  /«.  (See  Browne's  Mechanics,  p.  117.  Also  Magnus's 
Mechanics,  p.  157.) 

128.  The  Screw.— The  screw  consists  of  a  right  cir- 
cular cylinder,  on  the  convex  surface  of  which  there  is 
traced  a  uniform  projecting  thread,  abed  ....  inclined  at 
L  nstant  angle  to  straight  lines  parallel  to  the  axis  of  the 
cylinder.  The  path  of  the  thread 
may  be  traced  by  the  edge  AG  of 
an  inclined  plane,  ABC,  wrapped 
ronnd  the  cylinder;  the  base  of 
the  plane  corresponding  with  the 
circumference  of  the  cylinder,  and 
the  height  of  the  plane  with  the 
distance  between  the  threads  which 
is  called  the  pitch  of  the  screw. 
The  threads  may  be  rectangular  or 
triangular  in  section.  The  cylinder 
fits  into  a  blqck,  ou  the  inner  sur- 
face of  which  is  cut  a  groove  which  is  the  exact  counterpart 
of  the  thread.  The  block  in  which  the  groove  is  cut  is  often 
called  the  nut.  The  power  is  generally  applied  at  the  end  of 
a  lever  fixed  to  tho  centre  of  tlie  cylinder,  or  fixed  to  the  nut. 
It  is  evident  that  a  screw  never  requires  any  pressure  in  the 
direction  of  its  axis,  bnt  must  ^  made  to  revolve  only  ; 
and  this  can  be  done  by  a  force  acting  at  right  angles  to 
the  extremities  of  its  diameter,  or  its  diameter  produced. 

129.  The  Relation  between  the  Fewer  end  Jie 
Weight  in  the  Screw. — SuppiiM)  the  TH)wcr,  P,  to  act  in 
a  plane  por{)ondicular  to  the  axis  of  the  cylinder  and  at  the 
end  of  an  arm,  DE  =  a,  and  suppose  the  screw  to  have 
made  one  revolution,  the  power,  P,  will  have  moved 
through  the  circumference  of  which  a,  is  the  radius,  and 
tlie  work  done  by  P  will  be  Py.%ita.     During  the  same 


Fia.«7 


I 


^te 


MM 


ng  as  a  >  or 
k.180  Magnus's 


r  a  right  cir> 
bich  there  is 
.  iuclined  at 
le  axis  of  the 


J 


•7 

ooanterpart 
cat  is  often 
»t  the  eod  of 
i  to  the  nut. 
ssuro  in  tUa 
)volve  only  ; 
ht  angles  to 
troduced. 

and  Jh.9 

P,  to  act  in 
:*  and  at  the 
ew  to  have 
ave  moved 
radius,  and 
the  same 


I 


TBS  SCREW. 


205 


time  the  screw  will  have  moved  in  the  direction  of  its  axis 
t'.roug  1  the  distance,  AB  =  Sfrr  tan  a,  r  being  the  radius 
of  the  cylinder,  nnd  n  the  angle  which  the  thread  of  the 
screw  makes  with  its  base.  Then  as  this  is  the  direction  in 
which  the  resistance  is  encountered,  the  work  done  against 
the  resistance,  W,  is  Winr  tan  a.  Hence  if  no  work  is  lost 
the  equation  of  work  will  be 


P  X  2ffa  =  ff  X  2rrr  tan  «. 


(1) 


That  is  the  power  is  to  the  weight  as  the  pitch  of  the  screw 
is  to  the  circumference  described  by  the  power. 

If  there  is  friction  between  the  thread  and  the  groove,  let 
B  bo  the  normal  pressure  at  any  point,  p,  of  the  thread, 
and  nR  the  friction  at  this  poinf^,  iihen  the  work  done 
against  the  friction  in  one  revolution  is  ftI,R  2nr  sec  a,  I.R 
denoting  the  sum  of  the  normfd  reactions  at  all  points  of 
the  thread.     Hence  the  equation  of  work  is 


P  %na  =  W2irr  tan  «  -f  ^  2nr  sec  «li?. 


(2) 


But,  for  the  oqnilibrinm  of  the  screw,  resolving  parallel 
to  the  axis,  we  have 


therefore 


W  =  I.  (R  cos  a— nJt  sin  a), 
W 


ZR  = 


cc3a  —  (i  sm  a 


which  in  (2)  gives 


Pa  ^  WV  tan  «  H : — ; 

cos  a  —  /u  sm  n' 

or  Pa  =  Wr  tan  (a  -f  0), 

0  being  the  angle  of  friction. 


(3) 


306 


psoyr'a  D/jvmsgfmAL  sossw. 


129a.  Fxonj'u  DifEurential  8or«w.— If  h  denote  the 
pitch  of  a  screw  (1)  becomes 

aPira  =  Wh, 


which  expresses  the  relation  between  P  and  W,  when  fno- 
tion  is  neglected.  Therefore  the  mechanical  advantage  is 
gained  by  making  the  pitch  very  small.  In  some  cases, 
however,  it  is  desirable  that  the  screw  should  work  at  fair 
speed,  as  in  ordinary  bolts  and  nuts,  and  then  the  pitch 
must  not  be  too  small.  In  oases  where  the  screw  is  used 
specially  to  obtain  pressure,  as  in  screw-presses  for  cotton, 
etc.,  we  do  not  care  for  speed,  but  only  for  pressure.  But 
in  practice  it  is  impossible  to  get  the  pitch  very  small  from 
the  fact  that  *'  the  angle  of  inclination  is  very  flac,  the 
threads  run  so  near  each  other  as  to  be  too  weak,  in  which 
case  the  screw  is  apt  to  "  strip  its  thread,"  that  is,  to  tear 
bodily  out  of  the  hole,  leaving  the  thread  behind. 

Where  very  great  pressure  is  required  a  difierential  nut- 
bole  is  resorted  to.  Let  the  screw  work  in  two  blocks, 
A  and  B,  the  first  of 
which  is  fixed  and  the 
second  movable  along  a 
fixed  groove,  n ;  and  let 
A  be  the  pitch  of  the 
thread  which  works  in 


K  SOS      SSS| 


ns-M 


the  block,  A,  and  h'  the  pitch  of  the  thread  which  works 
in  the  block  B.  Then  one  revolution  of  the  screw  impresses 
two  opposite  motions  on  the  block,  B,  one  equal  to  h  in  the 
dii-ection  in  which  the  screw  advances,  and  the  other  equal 
to  /('  in  the  opposite  direction.  If  then  the  block,  B,  is 
connected  with  the  resistance  W,  we  have  by  the  principle 
of  work 

8P»T<i=:  »F (*-*'); 

and  the  requisite  power  will  be  diminished  by  diminishing 


h  denote 


W,  when  fric- 
advantage  is 
1  some  cases, 
work  at  fair 
iien  the  pitch 
I  screw  is  used 
es  for  cotton, 
iressnre.  But 
ty  small  from 
yery  flac,  the 
eak,  in  which 
hat  is,  to  tear 
ad. 

erential  nat- 
two  blocks, 


which  works 

ew  impresses 

to  h  in  the 

other  equal 

block,  B,  is 

he  principle 


diminishiDg 


aXAitPLKa. 


80t 


h  —  h\  By  means  of  this  torew  a  oompantiVBly  small 
pressnro  may  be  made  to  yield  a  preasnm  enormously 
greater  in  magnitude. 

EXAMPLES. 

1.  A  lever  10  ini.  long,  the  weight  of  which  is  4  lbs.,  and 
acts  at  its  middle  point,  balances  about  a  certain  point 
when  a  weight  of  6  lbs.  is  bung  from  one  end;  find  the 
point  Ant.  2  ins.  from  the  end  where  the  weight  is. 

2.  A  lever  weighing  8  lbs.  balances  at  a  point  3  ins.  from 
one  end  and  9  ins.  from  the  other.  Will  it  continue  to  bal- 
ance about  that  point  if  equal  weights  be  suspended  from 
the  extremities  ? 

3.  A  beam  whose  length  is  12  ft  balances  at  a  point  2  ft 
from  one  end  ;  but  if  a  weight  of  100  lbs.  be  hung  from  the 
other  end  it  balances  at  a  point  %  ft  ftam  th&t  end ;  find  the 
weight  of  the  beam.  Am.  2S  lbs. 

4.  A  lever  7  feet  long  is  6up|K>rted  in  «  horisontal  posi- 
tion by  props  placed  at  its  extremities  :  find  where  a  weight 
of  28  lbs.  must  be  placed  so  that  the  pressure  on  one  of  the 
props  may  be  8  lbs.  Ana.  Two  feet  from  the  e.nd. 

6.  Two  weights  of  12  lbs.  and  8  Ib&  respectively  at  the 
ends  of  a  horisontal  lever  10  feet  long  balance :  find  how 
far  the  fulcrum  ought  to  be  moved  for  the  weights  to  bal- 
ance when  each  is  increased  by  2  lbs.     Ann.  Two  inches. 

6.  A  lever  is  in  equilibrium  under  the  action  of  the  forces 
P  and  Q,  and  is  also  in  equilibrium  when  P  is  trebled  and 
Q  is  increased  by  6  lbs.:  find  the  magnitude  of  Q. 

Ans.  3  lbs. 

7(  In  a  lever  of  the  first  kind,  let  the  power  be  217  lbs., 
the  weight  726  lbs.,  and  the  angle  between  them  196°. 
Find  the  pressure  on  the  falcmm.  Am.  eS2.7  lbs. 


ao8 


BXAMPhEA 


8.  If  the  power  and  weight  in  a  straight  lever  of  the 
first  kind  be  17  lbs.  and  32  Ib&,  and  make  wi^'a  each  other 
an  angle  of  79° ;  find  the  pressure  on  the  fulcrum. 

An8.  39  lbs. 

9.  The  length  of  the  beam  of  a  false  balance  is  3  ft. 
9  ins.  A  body  placed  in  one  scale  balances  a  weight  of 
9  lbs.  in  the  other ;  but  when  placed  in  the  other  scale  it 
balanceb  4  lbs.;  required  the  true  weight,  W,  of  the  body 
and  the  lengt,hs,  a  and  b,  of  the  arms. 

Ans.    F  =  6  lbs.;  «  =  1  ft.  6  ins.;  i  =  2  ft  3  ins. 

10.  If  a  balance  be  false,  having  its  arms  in  the  ratio  of 
15  to  16,  find  how  much  per  lb.  a  customer  really  pays 
for  tea  which  is  sold  to  him  from  the  longer  arm  at  3s.  9d. 
per  lb.  ^n«.  4s.  per  lb. 

11.  A  straight  uniform  lever  whose  weight  is  50  lbs.  and 
length  6  feet,  rests  in  equilibrium  on  a  fulcrum  when  a 
weight  of  10  lbs.  is  suspended  from  one  extremity :  find  the 
position  of  the  fulcrum  and  the  pressure  on  it. 

Ana.  2^  ft  from  the  end  at  whioh  10  lbs.  is  suspended ; 
60  lbs. 


12.  On  one  arm  of  a  false  balance  a  body  weighs  11  lbs.; 
on  the  other  17  lbs.  3  02.;  what  is  the  true  weight  ? 

Ans.  13  lbs.  12  oz. 

13.  A  bent  lever  is  oompoeer  of  two  straight  uniform 
rods  of  the  same  length,  inclined  to  each  other  at  120°,  and 
the  fulcrum  is  at  the  point  of  intersection :  if  the  weight  of 
one  rod  be  double  that  of  the  other,  show  that  the  lever  will 
remain  at  rest  with  the  lighter  arm  horizontal. 

14.  A  uniform  lever,  /  feet  long,  has  a  weight  of  W  lbs., 
suspended  from  its  extremity ;  find  the  position  of  the  ful- 
crum when  the  long  end  of  the  lever  balances  the  short 


• 


t  lever  of  the 

ta  each  other 

rum. 

Ins.  39lbe. 

lance  is  3  ft. 
s  a  weight  of 
other  scale  it 
,  of  the  body 

2  ft  3  ins. 

in  the  ratio  of 
;r  really  pays 
arm  at  3s.  9d. 
4s.  per  lb. 

is  50  lbs.  and 
mm  when  a 
lity :  find  the 

i  suspended; 


ighs  11  lbs.; 

jht? 

lbs.  12  oz. 

fht  uniform 
at  120°,  and 
le  weight  of 
.he  lever  will 


of  W  lbs., 

of  the  ful- 

iis  the  short 


9 


SXAMPLKS. 


209 


end  with  the  weight  attached  to  it,  supposing  each  unit  of 
length  of  the  lever  to  be  w  lbs. 

■^nt.  ^T-n--,  J-  \  is  the  short  arm. 

15.  A  lever,  I  ft  long,  is  balanced  ,.  her  it 's  placed  upon 
a  prop  i  of  its  length  from  the  thick  end  ;  when  a  weight 
of  W  lbs.  is  suspendofl  from  the  small  end  the  prop  must 
be  shifted  j  ft.  towards  it  in  order  to  maintain  eqailibiium ; 
required  the  weig.t  of  the  lever.  Ana.  ^W. 

16.  A  lever,  I  ft.  long;  is  balanced  on  a  prop  by  a  weight 
of  W  lbs.;  first,  when  the  weight  is  suspended  from  the 
thick  end  the  prop  is  a  ft.  from  it;  secondly,  when  the 
weight  is  suspended  from  the  small  end  the  prop  is  b  ft. 
from  it ;  required  the  weight  of  the  lever. 

.         W{a  +  b)  ,. 
l  —  {a  -1,-0) 

17.  The  forces,  P  and  W,  act  at  the  arms,  a  and  b, 

respectively,  of  a  straight  lever.     When  P  and  W  make 

angles  of  30°  and  90°  with  the  lever,  show  that  when  equi- 

fibW 

librium  takes  place  P  = • 

a 

18.  Supposing  the  beam  of  a  false  balance  to  be  nniform, 
a  and  b  the  lengths  of  the  arms,  P  and  Q  the  apparent 
weights,  and  IV  the  true  weight ;  when  the  weight  of  the 
beam  is  taken  into  account  show  that 


a 
b 


P   -W 


19. 


W-Q 
If  a  be  the  length  of  the  short  arm  in  Ex.  14,  what 


must  be  the  length  of  the  whole  lever  when  equilibrium 

takes  place  P  /2aW 

Ana.  a  +  a/  —^^  +  aK 


w 


20.  A  man  whose  weight  is  140  lbs.  is  just  able  to  snp- 
port  a  weight  that  hangs  over  an  axle  of  6  ins.  radius,  by 


SIO 


MXAMTLMS. 


hanging  to  the  rope  that  passea  over  the  corresponding 
wheel,  the  diameter  of  which  is  4  ft;  find  the  weight  sup- 
ported. Ans.  560  lbs. 

21.  If  the  difference  between  the  diameter  of  a  wheel  and 
the  diameter  of  the  azte  be  aiz  times  the  radius  of  the  axie, 
find  the  greatest  weight  that  can  be  sustained  by  a  force  of 
60  lbs.  Ana.  240  lbs. 

22.  If  the  radius  of  the  wheel  is  three  times  that  of  the 
axle,  and  the  string  round  the  wheel  can  support  a  weight 
of  40  lbs.  onlj,  find  the  greatest  weight  that  can  be  lifted. 

Am.  120  lbs. 

23.  What  force  will  be  required  to  work  the  handle  of  a 
windlass,  the  resistaDce  to  be  overcome  being  1156  lbs.,  the 
radius  of  the  axle  being  six  ins.,  and  of  the  handle  2  ft. 
Sins.?  Ana.  216.75  lbs. 

24.  Sixteen  sailors,  exerting  each  a  force  of  29  lbs.,  push 
a  capstan  with  a  length  of  lever  equal  to  8  ft,  the  radius  of 
the  capstan  being  1  ft.  2  ins.  Find  the  resistance  which 
this  force  is  capable  of  sustaining. 

Ana.  1  ton  8  cwt.  1  qr.  17  lbs. 

25.  Supposing  them  to  have  wound  the  rope  round  the 
capstan,  so  that  it  doubles  back  on  itself,  the  radius  of  the 
axle  is  thus  increased  by  the  thickness  of  the  rope.  If  this 
be  2  ins.  how  much  will  the  power  of  the  instrument  be 
diminished.  '  Ana.  By  \,  or  12f  per  cent. 

26.  The  radios  of  the  axle  of  a  capstan  is  2  feet,  and  six 
men  push  each  with  a  force  of  one  cwt.  on  spokes  5  feet 
long ;  €nd  the  tension  they  wiU  be  able  to  prodnce  in  the 
rope  which  leaves  the  axle.  Ans,  m  cwt. 

27.  The  difference  of  the  diameters  of  a  wheel  and  axTe 
is  2  feet  6  inches ;  and  the  weight  is  equal  to  six  times  the 
power ;  find  the  radii  of  the  wheel  and  the  axle. 

Ant.  1%\xul;  3  ina. 


I 


0 


sorresponding 
s  weight  Bup- 
18.  560  lbs. 

tf  a  wheel  and 
IS  of  the  axle, 
by  a  force  of 
na.  240  lbs. 

E>8  that  of  the 
wrt  a  weight 
»n  be  lifted. 
w.  120  lbs. 

e  handle  of  a 
1156  lbs.,  the 
handle  2  ft. 
216.75  lbs. 

29  Iba,  push 
the  radins  of 
stance  which 

qr.  17  lbs. 

)e  round  the 
radius  of  the 
•ope.  M  this 
istmment  be 
per  cent. 

eet,  and  six 
spokes  5  feet 
oduce  in  the 
IS.  15  cwt. 

eel  and  axle 

six  times  the 

e. 

ns.;  dina. 


I  ' 


itAJtPLSa. 


an 


28.  If  the  rsdins  of  a  wheel  is  4  ft.,  and  of  the  a  tie 
8  ins.,  And  the  power  that  will  balance  a  weight  of 
500  lbs.,  the  thickness  of  the  rope  coiled  round  the  axle 
being  one  inch,  the  powe;  acting  without  a  rope. 

Ana.  88.54  lbs. 

29.  Two  given  weights,  P  and  Q,  hang  vertically  from 
two  points  in  the  rim  of  a  wheel  turning  on  an  axis; 
find  the  position  of  the  weights  when  equilibrium  takes 
plaoe,  supposing  the  angle  between  the  radii  drawn  to 
the  points  of  suspenrijn  to  be  90°,  and  that  6  is  the 
angle   which    the   radius,  drawn    to    /**&  point  of   sus- 


pension, makes  with  the  Tertical. 


Ana.  tan  d  = 


Q 


30.  What  weight  can  be  supported  on  a  plane  by  a  hori- 
zontal force  of  10  lbs.,  if  the  ratio  of  the  height  to  the  base 
isfP  Ana.  IS^lbs. 

31.  The  inclination  of  a  plane  is  30°,  and  a  weight  of 
10  lbs.  is  supported  on  it  by  a  string,  bearing  a  weight  at 
its  extremity,  which  passes  over  a  smooth  pulley  at  its 
summit ;  find  the  tension  in  the  string.  Ana.  5  lbs. 

32.  The  angle  of  a  plane  is  45° ;.  what  weight  can  be 
supported  on  it  by  a  horizontal  force  ot  3  lbs.,  and  a  force 
of  4  lbs.  parallel  to  the  '^Ivxe,  both  acting  together. 

Ana.  3  -H  4  V2  lbs. 

33.  A  body  is  supported  on  a  plane  by  a  force  parallel 
to  it  and  equal  to  |  of  the  weight  of  the  body ;  find  the 
ratio  of  the  height  to  the  base  of  the  plane. 

Ana.  1  :  2  -v/e. 

34.  One  of  the  longest  inclined  planes  in  the  world  is 
the  road  from  Lima  to  Oallao,  in  S.  America ;  it  is  8  miles 
long,  and  the  faU  is  511  ft.    Calculate  the  inclination. 

Am.  88'  27",  OT  1  yard  in  62. 


212 


SXAMPLBB. 


3? 


35.  If  the  force  required  to  draw  a  wagon  on  a  horizontal 
road  be  ^th  part  of  the  weight  of  the  wagon,  what  will  be 
the  force  required  tc  draw  it  up  a  hill,  the  elope  of  which 
is  1  in  43.  Ans.  -rrVitl'  pa^t  of  the  weight. 

36.  If  the  force  required  to  draw  a  train  of  cars  on  a 
IcTcl  railroad  be  yf^th  jmrt  of  the  load,  find  the  force 
required  to  draw  it  up  a  grade  uf  1  in  5(3. 

Ans.  xrS-jth  part  of  the  load. 

37.  What  force  is  required  (neglecting  friction)  to  roll  a 
ciisk  weigliing  964  lbs.  into  a  curt  3  St.  high,  by  means  of  a 
plank  14  ft  long  resting  against  the  cart. 

Arts.  The  force  must  exceed  206  lbs. 

38.  A  body  is  at  rest  on  a  smooth  inclined  plane  when 
the  power,  weight  and  normal  pressure  are  18,  26,  and 
12  lbs.  respectively ;  find  the  inclination,  a,  of  the  plane  to 
the  horizon,  and  the  angle,  6,  which  the  direction  of  the 
power  makes  with  the  plane. 

Ana.  a  =  37°  21'  26";  6  =  28°  46'  54". 

39.  If  the  power  which  will  support  a  weight  when  act- 
ing along  the  plane  be  half  that  which  will  do  so  acting 
horizontally,  find  the  inclination  of  the  plane.  Ana.  60°. 

40.  A  power  P  acting  along  a  plane  can  support  W;  and 
acting  horizontally  can  support  x ;  show  that 

P»=  »'»-««. 

41.  A  weight  W  would  be  supjwrted  by  a  power  P  act- 
ing horizontally,  or  by  a  power  Q  acting  parallel  t<j  the 
plane  \  show  that 

42.  The  base  of  an  iooliucd  plane  is  8  ft.,  the  height 
6  ft ,  and  TF  =  10  tons ;  required  P  and  the  uormal 
pressure,  N,  on  the  plane. 

Ana.  P  =  Q  tons ;  JV  =  8  tona. 


>n  a  horizontal 
,  what  will  be 
lope  of  which 
the  weight. 

I  of  cars  on  a 
Qnd  the  force 

of  the  load. 

;ion)  to  roll  a 
by  means  of  a 

eed  206  lbs. 

id  plane  when 
)  18,  26,  and 
i  the  plane  to 
•ection  of  the 

58°  46'  54". 

ht  when  act- 
do  so  acting 
Ana.  60°. 

)port  W;  and 


(owor  P  nct- 
rallel  t<j  the 


the  height 
the  uormal 

=  8  tons. 


SXAMPLEA 


Sid 


43.  A  weight  is  supported  on  an  inclined  plane  by  a 
force  whoso  direction  is  inclined  to  the  plane  at  an  angle 
of  30"  ;  wlieu  the  inclination  of  the  plane  to  the  horizon  ia 
30°,  show  that  IT  =  P  V'S- 

44.  A  man  weighing  150  Iba  raises  a  weight  of  4  cwt.  by 
a  system  of  four  movable  pulleys  arranged  according  to  the 
second  system ;  what  k  his  pressure  on  the  ground  ? 

Ans.  122  lbs. 

45.  What  power  will  be  required  in  the  second  system 
with  four  movable  pulleys  to  sustain  a  weight  of  17  tons 
12  cwt.  Ans.  1  ton  2  cwt. 

46.  Two  weights  hang  over  a  pulley  fixed  to  the  summit 
of  a  smooth  inclined  plane,  on  which  one  weight  is  sup- 
ported,  and  for  every  3  ins.  that  one  descends  the  other 
rises  2  ins.;  find  the  ratio  of  the  weights,  and  the  length 
of  the  plane,  the  height  being  18  ins.    Ana,  2  :  3  ;  27  ins. 

47.  U  W  =  336  Iba.  and  P  =  42  lbs.  in  a  combination 
of  pulleys  arranged  according  to  the  first  system,  how  many 
.novable  pulleys  are  there  ?  Ana.  4. 

48.  In  a  system  of  pulleys  of  the  third  kind  in  which 
there  are  4  cords  attached  to  the  weight,  determine  the 
weight,  W,  supported,  and  the  strain  on  the  fixed  pulley, 
the  power  being  100  lbs.,  and  th>.  weight,  W,  of  each 
pulley  6  lbs. 

Ans.  W  —  15P  •^-  llw  =  1666  lbs.;  Strain  =  16P  +  16u> 
=  1675  lbs. 

49.  In  a  system  of  pulleys  of  the  third  kind,  there  arc 
2  movable  pulleys,  each  weighing  2|  lbs.  What  power  is 
required  to  support  a  weight  of  6  cwt.  ?    Aiu.  94.67  Ibc. 

60.  Find  the  power  th»t  mill  support  a  weight  of  100  lbs. 
by  means  of  a  system  of  4  pulleys,  the  strings  being  all 
attached  to  the  weight,  and  each  pulley  weighing  1  lb. 

Ana.  5\^  lbs. 


814 


sxjZii'Lsa. 


i 


51.  The  cinmmfeMnoe  of  the  circle  oorrestwiulhig  to  tibe 
point  of  applicatioii  of  P  is  6  feet ;  find  how  many  tunui 
the  screw  must  make  on  a  cylinder  2  feet  long,  in  order 
that  IT  may  bo  equal  to  144P..  An».  48. 

62.  The  distance  between  two  consecutive  threads  of  a 
screw  is  a  quarter  of  an  inch,  and  the  length  of  the  power 
arm  is  5  feet;  find  what  weight  will  be  sustained  by  a 
power  of  1  lb.  Ann.  480Tr  Iba. 

53.  How  muiy  turns  must  be  giten  to  h  screw  formed 
upon  a  cylinder  whose  length  is  10  ins.,  and  circumference 
6  ins.,  that  a  power  of  %  ozs.  may  overcome  a  pressnre  of 
IOO0Z8.P  Ana.  100. 

54.  A  screw  is  made  to  '.evolve  by  a  force  of  2  lbs. 
applied  at  the  end  of  a  le.or  3.5  ft  long;  if  the  distance 
between  the  threads  be  \  in.,  what  pressure  can  be  pro* 
duced  r  Ant.  9  cwts.  1  qr.  20  lbs. 

56.  The  length  of  the  power-am  is  16  inches ;  find  the 
distance  between  two  consecutive  threads  of  the  screw, 
that  the  mechan   ol  advantage  may  be  30.      Ana.  it  ins. 

66.  A  weight  of  IF  pounds  is  suspended  f  •  +^i'  btflok 
of  a  single  movable  pulley,  and  the  end  f't  .:  ■  c**vii  hi 
which  the  power  acts,  is  fastened  at  the  d'!ut<.uii  <  h  ft 
from  the  fulcrum  of  a  horizontal  lever,  a  ft  long,  «.f  the 
second  kind ;  find  the  force,  P,  which  must  be  applied  per- 
p«Qdioakirly  at  th«  extremity  of  the  lever  to  initain  W. 

Ana.  P  =  -^-. 

57.  In  a  steelyard,  the  weight  of  the  bean  is  10  lbs.,  and 
the  distance  of  its  centre  of  gravity  from  the  fulcrum  it 
2  ins.,  find  where  a  weight  of  4  lbs.  most  be  phioed  to  bal- 
ance  it  Ana.  At  6  ini. 


'^nmmm^mmtmmgmm 


ipoo^hig  to  tile 
low  many  tnnifl 
i  long,  in  ordef 
Atu.  48. 

ve  threads  of  a 
h  of  the  power 
eostained  by  a 
rm.  4807rlba. 

a  Borew  fonned 

ciroiimferenoe 

e  a  pretsare  of 

Ana.  100. 

force  of  a  lbs. 
if  the  distance 
re  can  he  pro- 
1  qr.  20  Ibfl. 

ches;  find  the 
of  the  Bci-ew, 
Aru.  IT  ins. 

Ir*  fh'  block 
Ui  ■  c<»rd  in 

:i'     •  5    ft 

long,  *A  the 
>e  applied  per- 
luttain  W. 

P  =  I*. 
ia 

B 10  lbs.,  and 
le  fnlcnim  it 
ihioed  to  bal- 
At  Sim. 


MXAMPLM8. 


iUS 


68.  A  body  whose  weight  is  V2  lbs.,  is  placed  on  a  rongh 
plane  inclined  to  the  horizon  at  an  angle  of  46°.  The  co- 
efficient of  friction  being -p,  find  in  what  direction  a  force 

of  (V3  —  1)  lbs.  must  act  on  the  body  in  order  jnst  to 
support  it  Am.  At  an  angle  of  80°  to  the  plane. 

59.  A  rongh  piano  is  inclined  to  the  horizon  at  an  angle 
of  60°  ;  find  the  magnitni'e  and  the  Jireotion  of  the  least 
force  which  will  prevent  e  body  weighing  100  lbs.  fiom  slid- 
ing down  the  plane,  the  coefficient  of  friction  being  —  • 
Am.  60  lbs.  inclined  at  30°  to  the  plane. 


U^WIV'  ■«'««■ 


IlLJIBUIIIl    IMI^U. 


■W^ 


i 


CHAPTER    VIII. 

THE  FUNICULAR*  POLYGON— THE  CATENARY 
ATTRACTION. 

130.  fiqiiilibfilun  of  the  Fnniciilar  Poljrgon.— If  a 

cord  whose  weight  is  neglected,  is  saspended  From  two  fixed 
points,  A  and  B,  and  if  a  series  of  weights,  Pj,  P„  P„ 
etc.,  be  saspended  from  the  given  points  Q^,  Q„  Q^,  etc., 
the  cord  will,  when  in  equilibrinn:,  form  a  polygon  in  a 
vertical' plane,  which  is  called  the  Funicular  Polygon. 

Let  the  tensions  along 
the  successive  portions 
of  the  cord,  AQ I,  Qi Q^, 
QtQ»>  ®^  ^  respec- 
tively r,,  T„  r,,  etc., 
and  let  9,,  9„  9„  eta, 
be  the  inclinations  of 
these  portions  to  the 
horizon.  Then  Qi  is 
in  equilibrium  under  the  action  of  three  forces  viz.,  P,, 
acting  vertically,  7",,  die  tension  of  the  cord  AQ^,  and  7,, 
the  tension  of  Qi  Q,.     Resolving  these  forces  we  have, 

for  horizontal  foroes,        jT,  cos  0,  —  T,  cos  9,-0,  (1) 

for  vertical  forces,  P,  +  T,  sin  0,  —  T,  sin  9,  =  0,  (2) 

In  the  same  way  for  the  point  Q,  we  hare, 

for  horizontal  forces,        T,  cos  0,  —  T,  cos  9,  =  0,  (8) 

for  vertical  forces,  P,  +  T,  sin  9,  —  T,  sin  9,  =  0,  (4) 

•  The  term,  ^mtoulw,  bM  rafcrwM*  Alone  to  Um  oord,  ud  kaa  bo  ■«A«iilMd 
■ignttOHMe. 


Fia-M 


Mvomwii  .wTRf^f^w" 


I. 

CATENARY 

Poljrgon.— Ifa 

)d  from  two  fixed 
^tfl,  P„  P„  />„ 

^xj  Qt>  Qi,  etc., 
a  polygon  in  a 
%T  Polygon. 


i.n 

foHMS  via.,  P,, 
AQ„  and  T^, 
8  we  have. 

.0,  =. 

:0, 

(1) 

«»  = 

0, 

(2) 

».  = 

0, 

(8) 

(?,  = 

0, 

(4) 

hM  no  BcchMilMl 


MQUIUBRIUM  or  TBtt  WUmCULAS  POLTOON.      217 

Henoe  from  (1)  and  (8)  we  have 

T^  cog  0,  =  JT,  COB  ©,  =  J*,  cos  0,  =  etc. , 

that  is,  the  horitontal  components  of  the  tensions  iv  the  dif- 
ferent portions  of  the  cord  are  constant.  Lot  this  constant 
be  denoted  by  T;  then  we  have 


T,^ 


T» 


^.= 


COB  0, 


cos  tfj  *    ■*•  —  cos  e,  ' 

which  in  (2)  an ^  (4)  give 

P,  +  rtan  (J,  —  rtan  fi,  =  0, 

P,  +  r  tan  e,  —  rtan  8,  =  0, 

and  from  (5)  and  (6)  we  have 

P. 


;  etc.. 


(6) 


tan  e,  =  tan  e,  4-  -^ 


and 

Similarly 
and 


P, 

-f" 

P, 
T' 

p 

tan  0^  =  tan  fl,  4-  -^i 

etc.,  etc. 


tan  6,  =  tan  0,  + 


tan  0,  =  tan  e^  + 


(n 


If  we  suppose  the  weights  P|,  P,,  etc.,  each  equal  to  TT, 
(7)  becomes 


tan  ^i  —  tan  0g  =  tan  0,  ~  tan  9,  =  tan  0,  —  tan  04 

-  _  FT 


(8) 


Hence,  /A0  tangents  of  the  successive  inclinations  form  a 
series  in  Arithmetic  Progression.     In  the  figure  0,  =  0, 
10 


M^IWI    IWIKIH       II— I   IHWI>W 


818   coNSTancnotr  or  rat  rmncvLA»  polygon. 
tand,  =-^;    tan  0,  =  ^; 


T 


*»«»*•  =  -ST  ;  tan  e^  --fgr\  etc. 


(9) 


131.  To  Oonstrnct  the  Fanicnlar  Polygon  Trhen 
the  Horisontal  Progectioiui  of  the  euocesajve  Por- 
tions of  the  Cord  are  all  eqnaL— Let  ^,^4,  Q^q^,  q^q^, 
q,qi,  etc.,  be  all  of  constant  length  =  a,  and  let  Q^q^  ■=.  c. 
Then  since  by  (9)  of  Art. 
130,  the  tangents  of  9^,  d„ 
0,,  di,  etc.,  are  as  1,  2,  8, 
4,  etc.,  we  have 

Q^n  —  3Q,qt  =  3c;  etc. 

Hence,  taking  the  middle  point,  0,  of  the  honzontal 
portion,  QiQ^,  as  origin,  and  the  horizontal  and  vertical 
lines  through  it  as  axes  of  x  and  y,  the  co-ordiiateu  of  Q, 
are  (fa,  c) ;  those  of  Q,  are  ({a,  30) ;  those  of  Q^  .%-<^  (fa, 
6c),  and  those  of  the  nth  vertex  from  Q^  are  e  ^ idently 


o      2v  9i  Q' 


X  = 


2n  +  1  » («  +  1] 

— s a;   if  g=        A        'ft 


S 


Eliminating  n  from  these  equations  we  get 


0     ■'"4 


(1) 


which,  being  independent  of  ft,  is  ntisfled  by  all  the  ver- 
tices indifi'erently,  and  is  therefore  the  equation  of  a  oarre 
passing  through  all  the  vertices  of  the  polygon,  and 
denotes  a  parabola  whose  axis  is  the  vertical  line,  OY,  and 

whose  vertex  is  vertically  below  0  at  a  distance  =  x* 

The  shorter  the  distances  ^4^1,  QtQw  ®tc.,  the  more 
nearly  does  the  funiouiar  polygon  ooinoide  with  the  para- 
bolic curve. 


I'OLrodfr. 


tc. 


(9) 


olygon  T7hen 
tceMdive  Por- 

I  let  Q^q^  =  c. 


the  honzontal 
il  and  vertical 
rdiiatei)  of  Q, 

of  Qt  *"»  (K 
e  ^idently 


(1) 

tj  all  the  rer- 
ion  of  a  onrre 
polygon,  and 
ine,  OY,  and 
c 

8 
to.,  the  more 

th  the  para- 


|e  = 


O0M9  aVPFOMVmO  LOAA 


319 


:i32.  Cord  Bnppoiting  a  Load  TJnUemalj  Sis- 
tribnted  owr  th*  HoriscwtaL— If  the  number  cf  vertices 
of  the  polygon  be  very  great,  and  the  suBpended  weights  all 
equal  so  that  the  load  is  distribated  uniformly  along  the 
straight  line,  FE,  the  parabola  which  passes  through  all  the 
vertices,  virtually  coincides  with  the  cord  or  chain  forming 
the  polygon,  and  gives  the  figure  of  the  Suspension  Bridge. 
In  this  bridge  the  weights  suspended  from  the  successive 
portions  of  the  chain  are  the  weights  of  equal  portions  of 
the  flooring.  The  weight  of  the  chain  itself  and  the 
weights  of  the  sustaining  bars  are  neglected  in  comparison 
with  the  weight  of  flooring  and  the  load  which  it  carries. 


Fit.7l 


Let  the  span,  AB,  =  2a,  and  the  height,  OD,  =  h. 
Then  the  equation  of  the  parabola  referred  to  the  vertical 
and  horizontal  axes  of  x  and  y,  respectively,  through  0,  is 


y*  =  4mx, 


(1) 


4m  being  the  parameter. 

Because  the  load  between  O  and  A  is  uniformly  dis- 
tributed over  the  horizontal,  0£,  its  resultant  bisects  OB 
at  0;  therefore  the  tangents  at  A  and  0  intersect  at  0 
(Art  63). 

From  (1)  we  have 

^  _  8»  _  y. 


m 


I 


220 


CORD  aUPPORTINO  LOAD^ 


which  is  the  tangent  of  the  inolination  of  the  ottxve  at  any 
point  (a;,  y)  to  the  axis  of  x.  Hence  the  tangent  at  the 
p6int  of  support,  A,  makes  with  the  horizon  an  angle,  a, 

whose  tangent  is  — ,  which  also  is  evident  from  the  tri- 
angle ACE. 

Let  If  be  the  v^eight  on  the  cord  ;  then  ^  IF  is  the  weight 
on  OA,  and  therefore  is  the  yertical  tension,  V,  at  A.  Then 
the  three  forces  at  A  are  the  verticcJ  tension  V  ■=  \W,  the 
total  tension  at  the  end  of  the  cord,  acting  alonj^  who 
tangent  AG,  and  the  horizontal  tension,  T,  which  is  every- 
where the  same  (Art.  130).  Hence,  by  the  triangle  of 
forces  (Art  31)  these  forces  will  be  represented  by  tiie 
three  lines,  AE,  AC,  CE,  to  which  their  directions  are 
respectively  parallel ;  therefore  we  have  for  the  horizontal 
tension 

r  =  AE  cot  o  =  W^, 
and  the  total  tension  at  A  is 


4A 


EX  AMPLB, 


The  entire  load  on  the  cord  in  (Fig.  71)  is  320000  lbs.; 
the  span  is  150  ft.  and  the  height  is  15  ft.;  find  the  tension 
at  the  points  of  support  and  at  the  lowest  point  and  also  the 
inclination  of  the  curve  to  the  horizon  at  the  points  of 
support. 


tan  «  =  -—  =  .04 ; 
a 


a  ~  2V  48'. 


The  yertical  tension  at  each  point  of  support  is 
r  =  i  weight  =  160000  lbs. ; 


c 

t 

it 
t 

tl: 
cc 

B 

tb 
0 
ca 
or 
lei 
th 
ax 
lii 

P, 

hi 
th 

W( 


!:he  ottxve  at  any 

tangeut  at  the 

ZOQ  an  angle,  a, 

t  from  the  tri- 


r  W  is  the  weight 
F,  at  A.  Then 
nV=^W,  the 
;ting  aloni^  k.uo 
,  which  is  every- 
the  triangle  of 
resented  by  the 
•  directions  are 
r  the  horizontal 


is  320000  lbs.; 
tnd  the  tension 
Ut  and  also  the 

the  points  of 

48'. 
Irtif 


raw  COMMON  CATSXASr. 


221 


the  homontal  tension  is 

Tis  »r^  =  400000  lbs.; 

and  the  total  tension  at  one  end  is 

..        VV*+  T*  =  430813  lb& 

133.  The  Common  Catenary.— Its   Equation.— A 

catenary  is  the  curve  assumed  by  a  perfectly  flexible  cord 
when  its  ends  are  fastened  at  two  points,  A  and  B,  nearer 
together  than  the  length  of  the  cord.  When  the  cord  is  of 
constant  thickness  and  density,  t.  e.,  when  equal  portions  of 
it  are  equally  heavy,  the  carve  is  called  the  Common 
Catenary,  which  is  the  only  one  we  shall  consider. 

Let  A  and  B  be  the  fixed 
points  to  which  the  ends  of 
the  cord  are  attached ;  the 
cord  will  rest  in  a  vertical 
plane  passing  through  A  and 
B,  which  may  be  taken  to  be 
the  plane  of  the  paper  Let 
0  be  the  lowest  point  of  the 
catenary;  take  this  as  the 
origin  of  co-ordinates,  and 
let  the  horizontal  line 
through  0  be  taken  for  the 
axis  of  X,  and  the  vertical 
line  through  0  for  the  axis  of  y.  Let  {x,  y)  be  any  point, 
P,  in  the  curve  ;  denote  the  length  of  the  arc,  CP,  by  « ; 
let  c*  be  the  length  of  the  cord  whose  weight  is  equal  to 
the  tension  at  0 ;  and  T  the  length  of  the  cord  whose 
weight  is  equal  to  the  tension  at  P. 


V 

H' 

X     . 

\              " 

J 

*             4 

o 

A 

X 

T' 

n 

» 

*  Tb*  weif^t  of  a  nuU  of  ImgUi  of  tbe  conl  balog  here  Uken  m  the  nuit  o( 


TMa  COMMON  OATUNART. 


Then  the  arc,  CP,  after  it  has  assumed  its  permanent 
fonn  of  equilibrium,  may  be  considered  as  a  rigid  body 
kept  at  rest  by  three  forces  vis.:  (1)  T,  the  tension,  acting 
at  P  along  the  tangent,  (2)  c,  the  horizontal  tension  at  the 
lowest  point  C,  and  (3)  the  weight  of  the  cord,  CP,  acting 
vertically  downward,  and  denoted  by  a.  Draw  PT'  the 
tangent  at  P,  meeting  the  axis  of  y  at  T'.  Then  by  the 
triangle  of  forces  (Art  31),  these  forces  may  be  represented 
by  the  three  lines  PT,  NP,  T'N,  to  whici*  their  directions 
are  respectively  parallel.    Therefore 


or 


T'N  _  weight  of  CP 
~NP  ~  tension  at  C  * 

^  —  i. 
tke       0 


(«) 


Differentiating,  substituting  the  value  of  ds,  and  reducing, 
we  have 


d 


(I) 


y/Mtf 


e  ' 


Integrating,  and  remembering  that  when  x  =  0,  ^  =  0, 
we  obtain 

where  «  is  the  Naperian  base.    Solving  thif  equation  for 

2  =  »('--0'  (•> 


^,  we  obtain 


/ 


ed  ita  pennanent 
1  as  a  rigid  body 
he  tension,  acting 
tal  tennion  at  the 
cord,  CP,  acting 
Draw  PT'  the 
T'.  Then  by  the 
ay  be  represented 
:u  their  directions 


is.  and  reducing, 


.  =  0,g  =  0. 


X 

_    * 

c' 


ii«  eqaation  for 


THM  COMMON  CATMNAST. 


283 


and  by  integration,  observing  that  y  =  0  when  »  =  0, 
we  have 

9  =  1(^  +  0"')-^'  (8) 


which  is  the  equation  required.  We  may  simplify  this 
equation  by  moving  the  origin  to  the  point,  0,  at  a  dis- 
tance equal  to  c  below  0,  by  putting  y  —  c  for  y,  so  that 
(2)  becomes, 


=i(.%4 


(8) 


which  is  the  equation  of  the  catenary,  in  the  usual  form. 
The  horizontal  line  through  O  is  called  the  directrix*  of 
the  catenary,  and  0  is  called  the  origin. 

e  r 

Cob.  1.— To  find  the  length  of  the  arc,  AP,  we  have 

=  Y  I  +  i(e^  -  e'*}  dx,  from  (1), 

=  iK+e"7<^;  (4) 

.-.    «  =  |(<l-«"^  (6) 

the  constant  being  =  0,  since  when  x  =  0,  s  =  0. 

This  equa^ioQ  may  also  be  found  immediately  by  equa* 

dv 
ting  the  values  of  ^  in  (o)  and  (1). 

•  9m  Mo*'*  An«L  Mtebt.,  Tol.  I,  p.  tl«. 


8M 


THE  COMMON  CATENART. 


Cor.  2.— Since  c  =  OC  is  the  length  of  the  cord  whoae 
weight  is  equal  to  tha  tension  of  the  curve  at  the  lowest 
point,  0,  it  follows  that,  if  the  half,  BO,  of  the  curve  were 
removed,  and  a  cord  of  length  c,  and  of  the  same  thickness 
and  density  as  the  cord  of  the  curve,  were  joined  to  the 
arc  CP,  and  suspended  over  a  smooth  peg  at  C,  the  curve 
would  be  in  equilibrium. 

Cob.  3.— We  have  from  the  triangle,  PNT, 

tension  at  P  __  PT 
tension  at  C  ~  PiV  ' 


or 


!*_- *L  —  if 
c       dx       c 


m  (3)  and  (4), 


that  is,  the  tension  at  any  point  of  the  ^■'tenary  is  equal  to 
the  weight  of  a  portion  of  the  cord  whose  length  is  equal  to 
the  ordinate  at  that  point. 

Therefore  if  a  cord  of  constant  thickness  and  density 
hangs  freely  over  any  two  smooth  pegs,  the  vertical  por- 
tions which  hang  over  the  pegs,  must  each  terminate  on 
the  directrix  of  the  catenary. 


Cob.  4.— From  (3)  and  (5)  we  have 

y»  =  «»  +  (?, 

and  from  (6)  we  have 

dy 


(7) 


At  the  point,  P,  draw  the  ordinate,  PM,  and  from  M, 
the  foot  of  the  ordinate,  draw  the  perpendicular  MT.  Then 


pr  =  y  cos  MPT  =  y 


dy 


of  the  cord  whose 
mrve  at  the  lowest 

of  the  curve  were 
the  same  thickness 
vere  joined  to  the 
!g  at  C,  the  curve 

'NT, 


(4), 


'tenary  is  equal  to 
length  is  equal  to 

ness  and  density 

the  vertical  por- 

ach  terminate  on 


(6) 


(7) 

M,  and  from  M^ 
icularirr.  Then 


TBJS  COMMON  CATMUtABT. 

which  in  (7)  givQp 

PT  =:  9  =  the  OK,  CP,  (8) 

and  since  j/t  ^  PT»  +  TM*,  we  have  from  (6)  and  (8) 


TM=c. 


(9) 


Therefore  the  point,  T,  is  on  the  involute  of  the  oitenary 
which  originates  from  the  curve  at  C,  TM  ib  n  tangent  to 
this  involute,  and  TP,  the  tangent  to  the  catenary,  is 
normal  to  the  involute,  (Seo  alculus,  Art.  124).  As  TM 
is  the  tangent  to  this  last  v  nrve,  and  is  equal  to  the  con- 
stant quantity,  c,  the  involute  is  the  eq^itangential  Curve, 
ortractrix(SeeOalcaJus,  p.^67). 

By  means  of  (8)  and  (9)  we  may  construct  the  origin  &nA 
dtrecinx  of  the  catenary  as  follows  :  Oh  the  tangent  at  anv 
point,  P,  measure  off  a  Ungth,  PT,  equal  to  the  arc,  OP; 
at  T  erect  a  perpendicular,  TM,  to  the  tangent  meeting  the 
ordinate  of  P  at  M;  then  the  horizontal  line  through  M  is 
the  diredtnx,  and  its  intersection  with  the  axie  of  tlie  (mm 
is  the  origin. 

OoB.  6.— Combining  (%)  and  (5)  we  obtain 

(y  +  c)»  =  8»  +'  c». 


therefore 


«•  =  »»  +  2<?y. 


(10) 


The  catenary  possesses  many  interesting  geometric  and 
mecnanical  properties,  but  a  discussion  of  them  would 
carry  us  beyond  the  limits  of  this  treatise.  The  student 
who  wishes  to  pursue  ihe  subject  further,  is  referred  to 
Prices  Anal.  Mecha,  Vol.  I,  and  Miucbin's  Statics. 


m^tffiiiiifiiniifrSniiiif 


M 


I 


326 


BPSTBRWAL  8ESLL. 


133a.  Attraction  of  a  Sphwieal  SholL-By  the  law 

of  umverBal  gravitation  every  particle  of  matter  attracts 
every  other  particle  with  a  force  that  varies  diredly  as  the 
mass  of  the  attracting  particle,  and  inversely  as  the  square 
of  the  distance  between  the  particles. 

To  find  tJ^  temltant  attraction  of  a  spherical  shell  of 
um/orm  densttv  and  small  uniform  thickness,  on  a  par- 

(1)  Suppose  the  paiticle,  P, 
on  which  the  value  of  the 
attraction  is  required,  to  be 
outside  the  shell. 

Let  p  ana  *  be  the  density 
and  thickness  of  the  shell,  0 
its   centre    and   M  apj   particle  of  it.     Let  03f  =  a 
FM  =  r,  OP  =  c,  the  angle  MOP  =  0, 0  the  angle  which 
the  plane  MOP  makes  with  a  fixed  plane  through  OP      ^ 

^  ,  "  IL  .""^  **^  *^^  «'«'"«"*  »t  ^  (Art.  88)  is 
pka^,m9dBd^  The  attraction  of  the  whole  shell  L-ts 
along  OP)  the  attraction  of  the  elementary  mass  at  M  on 
P  in  the  direction  PM 

—  P^  <**  **»"  ^  M  d<p 

therefore  the  attraction. of  i/' on  P,  resolved  along  OP, 
_  pkcfi  Bin  6  de  d<p  c  — a  cos  0 


^ 

We  shall  eliminate  0  from  this  equation  by  means  of 
t*  =:  a»  +  d>  -  2ac  (m  e ; 
.*.    rdr  sz  ac  Bin  6  d9; 


(1) 


Shell— By  the  law 

of  matter  attracts 
lies  directly  an  the 
rseltf  aa  the  square 

I  spherical  shell  of 
tckness,  on  a  par- 


t.     Let   OM  =  o, 

;  0  the  angle  which 

I  through  OP. 
M  (Art.  88)  is 
whole  shell  sots 

ary  mass  at  M  on 


ed  along  OP, 
y  moaoa  of 


(1) 


ac 


substituting  these  values  in  (1),  the  attraction  ot  JM  on  P 
along  PO 

=  g.-(' +  -;-)**  w 

To  obtain  the  resultant  attraction  of  the  whole  shell,  we 
take  the  ^integral  between  the  limits  0  and  2n,  and  the 
r-integral  between  c  —  a  and  c  +  a. 
Hence  the  resultant  attraction  of  the  shell  on  P  along  PO 


npka 


1  + 


«»- 


^dr. 


^npkn*      mass  of  the  shell 


(3) 


Since  e  is  the  distance  of  the  point  P  from  the  centre  this 
shows  that  the  attraction  of  the  shell  on  tlie  paVticIo  at  P 
is  the  same  as  if  the  mass  of  the  shell  were  condensed  into 
its  centre. 

It  follows  iTom  this  that  a  sphere  which  is  either  homo- 
geueous  or  consists  of  concentric  spherical  shells  cf  uniform 
density,  attracts  the  particle  at  P  in  the  same  manner  as  if 
the  whole  mass  were  collootcd  at  its  centre. 

(2)  lict  the  particle,  P,  be  inside  the  sphere.  Then  wo 
proceed  exactly  as  beioro,  and  obtain  equation  (2),  which  is 
true  whether  the  particle  be  outside  or  inside  the  sphere ; 


■ 

h 


mmm 


MXAKPLMB. 

but  the  r-limits  in  thia  eaae  gre  «  —  c  and  a  +  c.    Hence 
from  (2)  we  have,  by  performing  the  ^integration, 

attraction  of  sheU  =  ^  J^  (l  -  ^-^^r)  ^^' 

=  !:^(2c-ac)  =  0. 

therefore  a  particle  within  the  ehell  is  equally  attracted  in 
every  diiection,  t. «.,  is  not  attracted  at  all. 

CoE.— 11  a  particle  be  inside  o  homogenons  sphere  at  Uhe 
distance  r  from  its  centre,  all  that  portion  of  the  sphere 
which  is  at  a  greater  distance  ftrom  the  centre  than  the 
particle  produces  no  effect  on  the  particle,  while  the  re- 
mainder of  the  sphere  attracts  the  particle  in  the  same 
manner  as  if  the  mass  of  the  remainder  were  all  collected 
at  the  centre  of  the  sphere.  Thiw  th.  attraction  of  the 
sphere  on  the  particle 


|7rpr» 


_  i-T—    or 


4Trpr 


Honoe,  toiihin  a  homogeneous  sphere  the  attraction  varies 
as  the  dittoes  from  the  centre. 

The  propositious  respecting  the  attraction  o.  »  uniform 
Bphericd  lihell  on  an  external  or  internal  particle  were 
given  by  Newton  (Principia,  Lib.  I,  Prop.  70,  71).  (See 
Todhunter's  Statics,  p.  276,  also  Pratt's  Mechs.,  p.  187, 
Price's  Anal.  Mechs.,  Vol.  I,  p.  «66,  Minohin's  Statics, 
p.  408). 

BXAM  PLBS. 

1.  The  span  Ali  =mO  foet,  and  CO  =  1600  feet,  And 
Uu'  length  of  the  curve,  CA,  the  height,  CH,  and  the 


nda  +  c.    Hence 
tegration, 


««-e» 


)dr. 


=  0, 


[aally  attracted  in 
JL 

noQB  sphere  at  Hbe 
tion  of  the  sphere 
9  centre  than  the 
icle,  while  the  re- 
ticle in  the  same 
were  all  collected 
attraction  of  the 


attraction  varies 

ioD  Oi  a  nniform 
lal  particle  were 
).  70,  71).  (See 
Mechs.,  p.  187, 
inohin's  Statics, 


=  1600  feet,  find 
t,  CH,  and  the 


MXAMPLMtH 


229 


iaoliaation,  B,  id  the  ewfe  te  the  horisoai  ii  eithsr  point  of 
siupension. 

(1)  Here  -  =  f,  and  <  ==:  3'7183d, 
c 


therefore 


^  =:  (a-71828>*  =  1-2840, 

and  0  •  =  (2. 71828p  =  0-  7788. 

Substitnting  these  values  in  (5)  we  get 

i$  =  800  X  0.5052  =  404-16. 
CA  =  404-16  feet. 


Hence 
(2) 


=  800  X  2-0628  —  1600 
=  60-24  r^ot 


(«) 


therefore 


tan 


«  =  ^?  =  t(^*---*). 


cte 


flrom  (1), 


=  0-2526, 
e  =  14°  11'. 


8  404-16 

Otherwise  tan  »  =  -,  from  (a),  =  -YaST  =  0-2526,  as 


before. 


1600 


2.  The  entire  load  on  the  cord  in  Fig.  71  is  160000  lbs., 
the  span  is  192  (t,  and  the  height  is  15  ft;  find  the  tension 
at  the  points  of  support,  and  also  the  tension  at  the  lowest 
point  Ana.  Tension  at  one  end  =  268208  lbs. 

Horizontal  tension  =  256000  "* 


Ii  : 


^lr* 


no 


MXAMPLXa. 


3.  A  chain,  AOB,  10  feet  long,  and  weighing  30  lbs.,  is 
aaspended  so  that  the  height,  CH,  =  4  feet ;  find  the 
horiaontal  tension,  and  the  inclination,  9,  of  the  chain  to 
the  horizon  at  the  points  of  support 

Am.  Horizontal  tension  =  3|  lbs.,  9  =  77°  19'. 

4.  A  chain  110  ft  long  is  suspended  from  two  points  in 
the  same  horizontal  plane,  108  ft.  apart;  show  that  the 
tension  at  the  lowest  point  is  1.477  times  the  weight  of  the 
chain  nearly. 


weighing  30  lbs.,  ig 
4  feet;  find  the 
0,  of  the  chain  to 

M.,  9  =  77°  19'. 

rom  two  points  in 

t;  show  that  the 

the  weight  of  the 


PART   II. 

KINEMATICS   (MOTION). 


CHAPTER    I. 

RECTILINEAR    MOTION. 

134.  Dsflnitioiia.— Velocity.  —  Kinematics  is  that 
branch  of  Djrnamics  which  treats  of  motion  without  refer- 
ence to  the  bodies  moved  or  the  foree*  producing  the  mo- 
tion (Art.  1).  Although  we  do  not  know  motion  as  free 
ttom  force  or  from  the  maiter  that  is  moved,  yet  there  are 
cases  in  which  it  is  advantageous  to  separate  the  ideas  of 
force,  matter,  and  motion,  and  to  study  motion  in  the 
abstract,  t.  «.,  without  any  reference  to  what  is  moving,  or 
the  cause  of  motion.  To  the  study  of  pure  motion,  then, 
we  devote  this  and  the  following  chapter. 

The  velocity  of  a  particle  has  been  defined  to  be  its  rate 
of  motion  (Art.  7).  The  formulae  for  uniform  and  variable 
velocities  are  those  which  were  deduced  in  Art.  8.  From 
(1)  and  (2)  of  that  Art.  we  have 


,1 


ds 

di' 


(1) 
(») 


in  which  v  is  the  velocity,  s  the  space,  and  t  the  time. 


■MtJMo. 


888 


EXAMPLB8. 


:X  AM  PI.E8. 


1.  A  body  moyes  at  the  rate  of  754  yar«l8  per  hour.  Find 
ihe  velocity  in  feet  per  second. 

Since  the  velocity  is  uniform  we  use  (1),  hence 

*'  =  7  =  H7^ STi  =  0.628  ft.  per  sec,  Ans. 

t        bU  X  oU 

2.  Find  the  position  of  a  particle  at  &  given  time,  /, 
when  the  velocity  varies  as  the  distance  from  a  given  point 
on  the  rdctiUnear  path. 

Here  the  velocity  being  variable  we  have  from  (2) 

(to       , 
*'  =  rf-<  =  *'' 

where  /b  is  a  constant ; 
d$ 


therefore 


=  kdt; 


logs  s=  kt  +  e, 


(1> 


where  c  is  an  arbitrary  coustant. 

Now  if  we  suppose  that  s^  is  the  distance  of  the  particle 
from  the  given  pomt  when  <  =  0  we  have  c  =  log  »,, 
which  in  (1)  gives 

log  —  =  ife/ ;  '  or    s  =  s,e^. 

3.  A  railway  train  tra'  Is  at  the  rate  of  40  mile«per 
hour ;  find  its  velocity  in  feet  per  second. 

Ans.  58.66  ft  per  ieoond. 

4.  A  train  takes  7  h.  31  m.  to  travel  300  miles  ;  find  its 
velocity.  Ans.  39.02  ft  per  eec. 

5.  If  «  =  4^,  find  the  velocity  at  the  end  of  five  seconds. 

Ans.  300  f  t^  per  sea 

6.  Find  the  position  of  the  particle  in  Ex.  2,  when  the 
Telocity  varies  as  th«  time.  Ans.  s  sx  s^  +  ^kfi. 


fl  per  hour.   Find 

,  bence 

per  sec,  Ans. 

I  giren  time,  t, 
•m  a  given  point 

I  from  (2) 


he, 


{1} 


of  the  particle 
m  c  =  log  «„ 


>f  40  milMper 

peraeoond. 

miloB ;  find  its 
ft  per  eec. 

of  five  seconds. 
it.  per  860. 

X.  2,  when  the 


AOCMLMSATnur  MMMO. 


233 


7.  Find  tb«  diertiance  the  particle  will  more  in  one 
minute,  when  the  Telocitj  is  10  ft.  at  the  end  of  one 
second  and  Ti<jries  as  the  time.  Aiu.  18000  ft. 

135.  Acceleration.  — Acceleration  has  been  defined  to 
be  the  rate  of  change  of  vOocity  (Art.  9).  It  is  a  wioeily 
increment.  The  formula  for  acceleration  are  £rom  (1),  (3), 
and  (3)  of  (Art.  10), 


f-dV 


/  = 


M"' 


(1) 
(2) 
(8) 


(1)  being  for  uniform,  and    (2)    and    (3)    for  variable, 

acceleration. 

If  the  velocity  decreases,  f  is  negative,  and  (2)  and  (3) 

become 

^^  -       /^.    *?  -  _  f . 
di~  ~^'    dfl  ~      ■'* 

and  the  velocity  and  time  are  inverse  functions  of  each 
other. 

136.  The  Relation  between  the  Space  and  Time 
when  the  Acceleration  =  0. 


Here  we  have 


*"-0 


so  that  if  V,  is  the  constant  velocity  we  have 

da 


s=» 


•  > 


«  =  «,<  +  «•» 


.1  r 


'f* 
.rt'2] 


I'  i 


tt4 


AOOELMnATIOIf  CONSTAltT. 


1< 

3 


i 


i 


in  which  9,  is  the  space  which  the  body  has  passed  over 
when  ^  =  0.  If  <  is  computed  from  the  time  the  body 
stairtj  from  rest,  then  a  =  v^t  The  student  will  observe 
that  this  is  a  case  of  uniform  velocity. 

137.  Th«  Relation  (1)  between  tiie  Space  and 
Time,  and  (2)  between  the  Space  and  Veloeity, 
when  the  Acceleration  ia  Oonatant 


(1)  Let  A  be  the  initial  position  of    cT 
the  particle  supposed  to   be  moving 


p 

Fig.73 

toward  the  right,  P  its  position  at  any  time,  t,  from  A,  v 
its  velooity  at  that  time,  and  /  the  constant  acceleration  c* 
its  velocity.  Take  any  fixed  point,  0,  in  the  line  of  motion 
as  origin,  and  let  OA  =  «, ;  OP  =  s.  Then  the  equation 
of  motion  ia 

•••        rf/    =/"   +   «• 

Suppose  the  velocity  of  the  particle,  at  the  point  A  to  be 
V,,  then  when  ^  =  0,  v  =  Vs;*  hence  e  ■=■  v„  and 


.«.    «  =  !//«  +  v^t  +  c'.. 
But  when  /  =  0,  «  =  «,;  hence  c'  =  «,,  and 
»  =  !//»  +  v,<  +  «„ 


(2) 


(3) 


Hence  if  a  particle  moves  from  rest  from  the  origin  O,  with 
a  constant  acceleration,  we  have 


*  OOM  MIM  ▼•loelty  and  ipaoe  rwpMtlTclr,  or  the  velocity  the  pwtide  lia% 
•nd  epMe  It  hM  moved  over  at  Uie  Inatant  ( begini  to  be  nekosed. 


T. 


iy  has  passed  over 
te  time  the  body 
dent  will  observe 


tlM  Space  and 
and  Velosity, 


Flo.73 

ime,  /,  from  A,  v 
it  acceleration  c* 
;he  line  of  motion 
hen  the  equation 


(1) 


point  A  to  be 
v„  and 


(8) 

and 

(8) 

origin  0,  with 


tdty  tbe  pwtlele  haM, 

MMd. 


ACCMJJISATION  VABtABLM. 


s:Aj,,-jr--';J;55.i; 


235 
(4) 


and  thus  the  space  described  varies  as  the  square  of  the 
time. 

(2)  From  (1)  we  have 

...    g  =  2/,  +  0. 

But  when  «  =  #„  t>  =  t», ;  hence  t)  =  t>,«  —  2/5jo,  and 

therefore 

v»  i:^  2/«  +  Vo"  -  2/8,.  (6) 

Equations  (2)  and  (3)  give  the  velocity  and  position  of  the 
particle  in  terras  of  t ;  and  (5)  gives  the  velocity  in  terms 
of  «. 

138.  When  the  Acceleration  Vaxiea  dirjcily  aa 
tiie  Time  from  a  State  of  Rest,  find  the  Velocit7 
and  Space  at  ttie  end  of  the  Time  t. 


Here 


'    '         dt 

where  «,  is  the  initial  velocity  ; 

.-.    »  r=  !«<•  +  V, 

the  initial  space  being  0  since  /  is  estimated  from  rest. 

139  When  the  Acceleration  ▼ariea  direotiiy  aa 
the  IMatance  ftam  a  giwen  Point  bt  the  line  of  Mo- 
tion, and  ia  negattve,  find  the  Relation  between 
the  Space  and  Time. 


1  :|;:vj 


-'.  ?■ 


m 


Jl 


itL.y.iy«iWiiiJiuii,v 

286 
Here 


J 


BXAMPLBS. 


==-ks; 


by  calling  «,  the  valae  of  $  when  the  particle  is  at  rest 

da 


V^o*  -  «• 


ifcirf/. 


the  negative  sign  being  taken  since  the  particle  is  moving 
towards  the  origin ; 

.  • .    cos-'  —  =  kU, 


if  «  =  «,  when  /  =  0 ; 


i^t. 


EXAMPLES. 

1.  A  body  commences  to  move  with  a  velocity  of  30  ft. 
per  sec,  and  its  velocity  is  increased  in  each  second  by 
10  ft.    Find  the  spoce  de«cribed  in  5  seconds. 

Here  /  =  10,  v,  =  30,  s^  =  0,  and  /  =  6,  thei^fore 
from  (2)  we  have 

«  =  |.10.a6  +  30.5  =  275,  Ans. 

2.  A  body  starting  with  a  velcoity  of  10  ft  per  sec,  and 
moving  with  a  constant  acceleration,  describes  90  ft.  in 
4  sees.;  find  the  acceleration.  Ana.  6^  ft.  per  sec. 

3.  Find  the  velocity  of  a  body  which  starting  fipom  reit 
with  an  acceleration  of  10  ft  per  sec.,  has  described  a  space 
of  20  a  Ana.  Wtt, 


;le  ifl  at  rest 


rticle  is  moying 


locity  of  30  ft 
lach  second  by 

:  5,  therefore 


Ins. 

per  sea,  and 
jbea  90  ft.  in 
1^  ft  per  sec. 

ing  from  rest 
bribed  a  space 
Ana.  30  ft 


FALLtJtB  BODina. 


4.  Throngh  what  space  mast  a  body  pan  nnder  an  accel- 
eration of  6  ft  per  sec,  so  that  its  velocity  may  increase 
from  10  ft  to  20  ft  per  sec.  ?  Ans.  30  ft 

5.  In  what  time  will  a  body  moving*  with  an  cxxselera' 
tion  of  25  ft  per  sec.,  acquire  a  velocity  of  1000  ft  per 
second?  '  Ana.  40  sees. 

6.  A  body  starting  flrort  rest  has  been  moving  for  5  min- 
utes, and  has  acqaired  a  velocity  of  30  miles  an  hour; 
what  is  the  acceleration  in  feet  per  second  ? 

Ana.  W  ft.  per  sec. 

7.  If  a  body  moves  from  rest  with  an  acceleration  of  {  ft 
per  sec,  how  long  mast  it  move  to  acquire  a  velocity  of 


40  miles  an  hour? 


Ans.  88  sees. 


140.  Equations  of  Motion  for  Falling  Bodies.— 

The  most  important  case  of  the  motion  of  a  particle  with  a 
constant  uccoleration  in  its  line  of  motion  is  that  of  a  body 
moving  under  the  action  of  gravity,  which  for  smtdl  dis- 
tances above  the  earth's  surface  may  be  considered  constant. 
When  a  body  is  allowed  to  fall  freely,  it  is  found  to  acquire 
a  velocity  of  about  32.2  feet  per  second  during  every  second 
of  its  motion,  so  that  it  moves  with  an  acceleration  of  32.2 
feet  per  second  (Art.  21).  This  acceleration  is  less  at  the 
summit  of  a  high  mountain  than  near  the  surface  of  the 
earth  ;  and  less  at  the  equator  than  in  the  neighborhood  of 
the  poles ;  i.  e.,  the  velocity  which  a  body  acquires  in  falling 
freely  for  one  nocond  varies  with  the  latitude  of  the  place, 
and  vrith  its  aliitueh  above  the  sea  level ;  but  is  independ- 
ent of  the  site  of  the  body  and  of  its  masa.  Practically, 
however,  bodies  do  not  fall  J>eely,  as  the  resistance  of  the 
air  opposes  their  motion,  and  therefore  in  practical  cases  at 
high  speed  (0.  g.,  in  artillery)  the  resistance  of  the  air  must 
be  taken  into  account    But  at  present  we  shall  neglect 

*  In  each  cate  tbe  body  ia  «nppoMd  to  etart  from  rect  nnleH  otherwise  stated. 


iij 


! 

II 


238 


FALLTNO  BODIES. 


this  resistance,  and  consider  the  bodies  ad  moving  in  vacuo 
under  the  action  of  gravity,  t.  e.,  with  a  constant  accelera- 
tion of  about  32.2  feet  per  second. 

As  neither  the  substance  of  the  body  nor  the  cause  of 
the  motion  needs  to  be  taken  into  consideration,  all  prob- 
lems relating  to  falling  bodies  may  be  reganled  as  cases  of 
accelerated  motion,  and  treated  from  purely  geometric 
considerations.  Therefore  if  we  denote  the  acceleration  by 
g,  as  in  Art.  23,  and  consider  the  particle  in  Art.  137  to  be 
moving  vertically  downwards,  then  (2),  (3),  (5)  of  Art  137 
become,  by  substituting  jr  for/, 


s  =  yp  +  vj  +  So, 
v*  =  2flf«  4-  Vo'  —  2jrso» 


(A) 


s  being  measured  as  before  from  a  fixed  point,  0,  in  the 
line  of  motion. 

Suppose  the  particle  to  be  projected  downward  from  O, 
then  A  commences  with  0  and  «,  =  0.  Hence  (A)  be- 
comes 

V  =zgt  +  v^,  (1) 


8  =yi*  +  v^t, 
i^  =  2g&  +  «,». 


(2) 
(3) 


As  a  particular  case  suppose  the  particle  to  bo  dropped 
from  rest  at  0  (Fig.  71).  Thro  A  coincides  with  0,  and 
s,  =  0,  Vj  =  0.    Hence  eqiov-ions  (A)  become 


v  =  gt. 

(4) 

» =  igi', 

(6) 

tf>  =  2ga. 

(6) 

WsmKiasmr- 


'jn-rn 


ovmg  m  vacuo 
istant  accelera- 

or  the  cause  of 
ation,  all  prob- 
rckd  as  cases  of 
rely  geometric 
acceleration  by 
I  Art.  137  to  be 
(5)  of  Art.  137 


(A) 


oint,  0,  in  the 

iward  from  O, 
Hence  (A)  be- 

(1) 

(«) 

(8) 

to  be  dropped 
with  0,  and 
me 


(») 
(«) 


PARTIOLX  PROJECTED  UPWARDS. 


239 


141.  When  the  Particla  im  Projected  Vextioally 
TTpwards. — Here  if  we  measure  s  upwards  from  the  point 
of  projection,  0,  the  acceleration  tends  to  diminish  the 
space  and  therefore  the  acceleration  is  negative,  and  the 
equation  of  motion  is  (Art  135) 


=  -9- 


In  other  respects  the  solution  is  the  same.  Taking 
therefore  «,  =  0  in  (A)  and  changing  the  sign  of  g,*  we 
obtain 

t>  =  Vo  —  gt,  (1) 


t)»  =  Vo»  -  ^gif. 


(2) 
(3) 


Gob.  1. — The  time  during  which  a  particle  rises  when 
projected  vertically  upwards. 

When  the  particle  reaches  its  highest  point,  its  velocity 
is  zero.  If  therefore  we  put  v  =  0  in  (1),  the  corresiwnd- 
ing  value  of  t  will  be  the  time  of  the  particle  ascending  to  a 
state  of  rest. 

9 

Cor.  2. — The  time  qf  flight  hefwe  returning  to  the  dart- 
ing point. 

From  (3)  we  have  the  distance  of  the  particle  from  the 
starting  point  after  ^  seconds,  when  projected  vertically 
upwards  with  the  velocity  v,.  Now  when  the  particle  has 
risen  to  its  maximum  height  and  returned  to  the  point  of 
projection,  «  =  0.  If,  therefore,  we  put «  =  0  in  (2),  and 
solve  for  /,  we  shall  get  the  time  of  flight.     Therefore, 

•  g  in  poaltlve  or  mgaUye  Moordin^  aa  the  particle  is  deaoendlng  or  aa- 
cendingr. 


■^  r  i 


240  FABTICLM  PROJSCTKD   UPWAMl^H. 


which  givM 


/  =  0,    or 


tv. 


I 


The  first  yaluo  of  t  shows  the  time  before  the  part'cle 

starts,  the  latter  shows  tine  time  when  it  has  returned. 

2r 
Hence,  the  whole  time  of  flight  is  -^,  which  is  just  double 

the  time  of  rising  (Cor.  1) ;  that  is,  the  time  <if  riniiig  equals 
the  time  of  falling. 

The  final  velocity,  by  (1)  of  Art.  140,  =  y/  =  ^  x  *^ 

(Cor.  1)  =  Wo  ;  hence  a  body  returns  to  any  point  in  its 
path  with  the  same  velocity  at  which  it  left  it.  In  other 
words,  a  body  passes  each  point  in  its  path  with  the  same 
velocity,  whether  rising  or  falling,  since  the  velocity  at  any 
point  may  be  considered  as  a  velocity  of  projection. 

• 

Cob.  ^.—The  greatest  height  to  which  the  partich  tuiU 
riM, 

At  the  summit  w  =  0,  and  the  corresponding  value  of  « 
will  be  the  greatest  height  to  which  the  particle  will  rise  ; 
when  V  =  0,  (3)  becomes 

o.t  — 


Co*.  4.— Since  v^*  —  2g8,  where  «  is  the  height  frum 
which  a  body  fiills  to  gpin  the  velocity  w„  it  follows  that  » 
body  will  rise  through  the  same  space  in  losing  a  velocity 
V,  as  it  would  fall  through  to  gain  it 


(fore  the  part^'cle 
it  has  returned. 

cb  is  just  double 

» <^  rining  equaU 


any  point  in  its 
3ft  it.  In  other 
t  with  the  same 
n  velocity  at  any 
ectiou. 


Ihe  particle  toiu 


ding  value  of  « 
ftiole  will  rise ; 


le  height  frum 
follows  that  a 
ling  a  velocity 


1.  A  body  projected  vertically  downwards  with  a  velocity 
of  20  ft.  a  see.  from  the  top  of  a  tower,  reaches  the  ground 
in  2.5  sees.;  find  the  height  of  the  tower. 

Here  t  =  ^,  and  v^  ~%0',  assume  g  =  32.  Then 
from  [i)  of  Art.  140  we  havo 

,  =  it^UL  +  20  X  f  =  150  ft 

2.  A  body  is  projected  vertically  upwards  with  a  velocity 
of  200  ft.  per  second ;  find  the  velocity  with  which  it  will 
pass  a  point  100  ft.  above  the  point  of  projection. 

Here  t>,  =  200,  «  =  100 ;  therefore  from  (3)  we  have 
»8  =  40000  -  6400  =  33600  ; 
.-.    v  =  40'v/2T. 

3.  A  man  is  ascending  in  a  balloon  with  a  uniform 
velocity  of  20  ft.  per  sec.,  when  he  drops  a  stone  which 
reaches  the  ground  in  4  sees.;  find  the  height  of  the 
balloon. 

Here  v,  =  20,  and  f  :=  4 ;  therefoi-e  from  (2)  we  have, 

after  changing  the  sign  of  the  second  mcrabor  to  make  the 

result  positive. 

fl  =  -  (80  -  256)  =  176, 

which  was  the  height  of  the  balloon. 

4.  A  body  is  projected  upwards  with  a  velocity  of  80  ft. ; 
after  what  time  will  it  return  to  the  band  ? 

Ana.  5  second*. 

8.  With  what  velocity  must  a  body  be  projected  ver- 
tically upwards  that  it  may  rise  40  ft.  P 

Atts.  10  VlO  ft.  per  SHJ. 

a 


,?     . !' 


-.^>J^.,-Jtf.,........:.|^.«^ 


'mmmamm 


242 


COMPOSITION  or  VBLOCITISa. 


6.  A  body  projected  vertioallj  upwards  paBses  a  oertain 
point  with  a  velocity  of  80  fl.  per  sec. ;  how  much  higher 
will  it  ascend  P  Ans.  ICJ  ft. 

7.  Two  balls  are  dropped  from  the  top  of  a  tower,  one  of 
them  3  sees,  before  the  other  ;  how  far  will  they  be  apart 


5  sees,  after  the  first  was  let  £eiI1  ? 


Am.  336  ft. 


8.  If  a  body  after  having  fallen  for  '3  sees.  breakF  a  pane 
of  glass  and  thereby  loses  one-third  of  its  velocity,  find  the 
entire  space  through  which  it  will  have  fallen  in  4  sees. 

Ans.  224  ft. 

142.  Composition  of  Velocities.— (1)  From  the  Par- 
allelogram of  Velocities,  (Art.  29,  Fig.  2),  we  see  that  if  AB 
represents  in  magnitude  and  direction  the  space  which 
would  be  described  in  one  second  by  a  particle  moving  with 
a  given  velocity,  and  AC  represents  in  magnitude  and 
direction  the  space  which  would  be  described  in  one  second 
by  another  particle  moving  with  its  velocity,  then  AD,  the 
diagonai  of  the  parallelogram,  reprenenta  the  resultant 
velocity  in  magnitude  and  direction. 

(2)  Hence  the  resultant  of  any  two  velocities,  as  AB,  BD, 
(Fig.  2),  is  a  velocity  represented  by  the  third  side,  DA,  of 
the  triangle  ABD;  and  if  a  point  have  simultaneously, 
velocities  represented  by  AB,  BC,  CA,  the  sides  of  a  trian- 
gle, taken  in  the  name  o-der,  it  is  at  rest. 

The  linos  which  are  taken  to  represent  any  given  forces 
may  clearly  be  taken  to  represent  the  velocities  which 
measure  these  forces  (Art,  19),  therefore  from  the  Polygon 
and  Parallelopiped  of  Forces  the  Polygon  and  ParaUel- 
ojtiped  of  Velocities  follow. 

(3)  Hence,  if  any  number  of  velocities  be  represented  in 
magnitude  and  direction  by  the  sides  of  a  closed  polygon, 
taken  all  in  the  same  order,  the  resultant  is  tsro. 

(4)  Also,  if  three  velocities  be  represented  in  magnitude 


passes  a  certain 
w  much  higher 
Ana.  ICt)  ft. 

'  a  tower,  one  of 
I  they  be  apart 
Atu.  336  ft. 

.  breake  a  pane 
ilocity,  find  the 
m  iu  4  sees. 
Ana.  224  ft. 

Prom  the  Par- 
)  see  that  if  AB 
le  space  which 
3le  moving  with 
magnitude  and 
1  in  one  second 
then  AD,  the 
the    reaullant 

es,  aa  AB,  BD, 
side,  DA,  of 
nmullaneoualy, 
ka  of  a  triau- 

ly  given  forces 
tlocitiea  which 
the  Polygon 
and  Parallel- 

repreaented  in 
ihaed  polygon, 
ro. 

in  magnitude 


■W'^^>:iS?^^J!jf»'A*^*fi9ft!j¥JV»>:|^ 


BEaOLUnON  OF  VBLOCPtlKS. 


243 


and  direction  by  the  thre*  edges  of  a  parallelepiped,  the  re- 
aultant  velocity  will  be  repreaented  by  the  diagonal. 

(5)  When  there  are  two  velocities  or  three  velocities  in 
two  or  in  three  rectangular  directions,  the  resultant  is  the 
square    root   of   the    sum    of   their    squares.      Thus,  if 

-£,  ^-,  ^j,  -^,  are  the  velocities  of  the  moving  point  and 

its  components  parallel  to  the  axes,  we  have  from  (2)  of 
Art.  30, 


and  from  (1)  of  Art.  34, 


(1) 


da 
dt 


=V6?)V(I)^(I)*       <') 


143.  Resolution  of  Velooitie>.— Af  the  diagonal  of 
the  parallelogram  (Fig.  2),  whose  sides  re])re8ent  the  com- 
ponent velocities  was  found  to  represent  the  resultant 
velocity,  so  any  velocity,  represented  by  a  given  straight 
line,  may  be  resolvrd  into  component  velocities  represented 
by  the  sides  of  the  parallelogram  of  which  the  given  lino 
\  the  di'\gonaL 

It  will  b<    rtasily  seen  that  (2)  of  Art.  134  is  equally 

applicable  whether  iiie  point  bo  considered  as  moving  in  a 

straiglit  line  or  in  a  curved  line ;  but  since  in  the  lalter 

ase  the  direction  of  motion  continually  changes,  the  mere 

liount  of  the  velocity  is  not  sufficient  to  describe  the 
ii  'tion  completely,  so  it  will  be  necessary  to  know  at  every 
instant  the  direction,  &b  well  as  the  magnitude,  of  the  point's 
velocity.  In  such  oasec  as  thie  Ihe  method  commonly  cm- 
ployed,  whether  we  deal  with  velocities  or  accelerations, 
consist.)  mainly  in  studying,  not  the  velocity  or  acceleration, 
directly,  but  its  components  parallel  to  any  three  assumed 
rectangular  axes.    If  the  {mrticle  be  at  the  point  (x,  y,  «), 


'  ,  i 


n 


1  ■ 


a:    ) 


"  f  1      i     1- 

i  I 


I'- 


Jj!l 

if} 

.     i 

-J 

I 


tfiiMMn 


^jMW  imitn  iili  Hiiii'ftyiJiiMiMHiiiri  ■  n 


iii_rnm-i  Ti-Tn-iiriT-TT'iir""'V  iiirir-rfT"" 


i- 


2U 


BXAMPLS9. 


at  the  time  /,  and  if  we  denote  its  relocitiet  parallel 
respectively  to  the  three  axes  by  «,  v,  w,  we  hare 

dx  dy  dz 


dt 


dt 


dt 


Denoting  by  v  the  velocity  of  the  moving  particle  along 
the  curve  at  the  time  t,  we  have  as  above 


and  if  o,  /3,  y  be  the  angles  which  the  direction  of  motion 
along  the  curve  makes  with  the  axes,  we  have,  as  in  (2)  of 

(Art  34),  • 

dx       ds 

^  =s  ^  oca  a  =  V  008  «  =  u; 

dt       da 

^  =  J-.  cos  y  =  »  cos  y  =  «>. 

m       dt 

.    ,  .     d!*    rfv    rf'    •     1.     u 

Hence  c«*ch  of  the  components  ^,  ^,  j^  is  to  be 

found  from  the  whole  velocity  by  retolving  the  velocity, 
I.  e.,  by  multiplying  the  velocity  by  coeim  of  the  angle 
between  the  direction  of  motion  and  that  of  the  compo- 
nent. 

EXAMPLES. 

1.  A  body  moves  under  the  influence  of  two  velocities, 
at  right  angles  to  each  other,  equal  respectively  to  17.14  ft. 
and   13.11  ft.  per  second.     Find  the  magnitude  of  tb" 
resultant  motion,  and  the  angles  into  which  it  divides 
right  angle. 

Ans.  81.679  ft.  per  sec. ;  37°  «ft'  and  68° 


ities   parallel 


ITC 


•article  along 


I 


(1) 


)n  of  motion 
>  as  in  (2)  of 


h 

if 


is  to  be 


the  velocity, 

of  the  angle 

the  compo- 


▼elocities, 
to  17.14  ft. 
ode  of  tb" 
divide* 

dSJ" 


MOnON  ON  AH  tlfCUNSD  PLANB, 


%.  A  ship  Bails  due  north  at  the  rate  of  4  knots  per 
hour,  and  a  ball  is  rolled  towards  the  east,  across  her  deck, 
at  right  angles  to  her  motion  at  the  rate  of  10  ft.  per 
second.  Find  the  magnitude  and  directicoi  of  the  real 
motion  of  tlie  ball. 

Am.  12.07  ft  per  soc.;  and  N.  66°  E. 

3.  A  boat  moves  N.  30°  E.,  at  the  rate  of  6  miles  per 
hour.    Find  its  rate  of  motion  northerly  and  easterly. 

Ans.  5.2  miles  per  hour  north,  and  3  miles  per  hour 
east. 

144.  Motion  on  an  ZncUnod  Plane.— By  an  exten- 
sion of  the  equations  of  Art.  140,  we  may  treat  the  case  of 
a  pai-tiole  sliding  from  rest  down  a  smooth  inclined  plane. 
As  tb>is  is  a  very  simple  case  in  which  an  acceleration  is 
resolved,  it  is  convenient  to  treat  of  it  in  this  part  of  our 
work ;  yet  as  it  properly  belongs  to  the  theory  of  con- 
strained motion,  we  are  unable  to  give  a  complete  solution 
of  it,  until  the  principles  of  such  motion  have  been  ex- 
plained iu  a  future  chapter. 

Let  P  bo  the  position  of  the  particle  at 
any  time,  t,  ou  the  inclined  plane  OA,  OP 
sr  «,  its  distance  from  a  fixed  point,  O,  iu 
the  line  of  motion,  and  let  «  be  the  inclina- 
tion of  OA  to  the  horizontal  line  AB.  Let 
Pi  represent  g,  the  vertical  acceleration  with  rfj-M 

which  the  body  would  move  if  free  to  fall.  Resolve  this 
into  two  components,  Po  =  ^  sin  a  along,  and  Vc  —  g 
cos  a  perpendicular  to  OA.  The  component  g  cos  a  pro- 
duces pressure  on  the  plane,  but  does  not  affect  the  motion. 
The  only  acceleration  down  the  plane  is  that  component  of 
the  whole  acceleration  which  is  parallel  to  the  plane,  vie., 
g  sin  «c.    The  equation  of  motion,  therefore,  if 


^  =  ^  sin  «, 


(1) 


— f- 


j».t* 


^^..^^■liiHrWBIIilllttfi 


MIMIMM 


240 


DBCOENT  DOWN  CHORDS  OF  A  OIBCLX. 


the  Bolation  cf  which,  aag  nintt  is  constant,  is  incladed  in 
that  of  Art.  140;  and  all  the  resalts  for  particles  moving 
vertically  aa  given  in  Arts.  140  and  141  will  be  made  to 
apply  to  (1)  by  writing  g  sin  et  for  g.  Thns,  if  the  particle 
be  projected  down  or  up  the  plane,  we  get  from  (1),  (2),  (3) 
of  Arts.  140  and  141,  by  this  means 


V  =  Vo  ±  jf  sin  u't, 
s  =  Vft  ±  \g  sin  a- 1, 
V*  =  v,»  ±  2^  sin  «•«, 


(3) 
(4) 


in  which  the  -f  or  —  sign  is  to  be  taken  according  as  the 
body  is  projected  down  or  up  the  plane. 

If  the  particle  starts  from  rest  from  0,  we  get  from  (4), 
(5),  (6)  of  Art.  140 

V  =  ^  sin  a*  t,  (6) 


a  =  4jr  sin  a-  ^, 
»»  =  2jf  sin  «•«. 


(8) 
(7) 


CoE.  1. — Tk«  velocity  acquired  by  a  particle  in  falling 
down  a  given  inclined  plane. 

Draw  PO  parallel  to  AB  (Fig.  74),  then  if  »  be  the 
velocity  at  P,  wo  have  from  (7) 

1^  =  2g  sin  a*« 

Hence,  from  (6)  of  Art.  140  the  velocity  is  the  same  at  P 
as  if  the  particle  had  fullen  throngh  the  vertical  space  OC  ; 
that  is,  tlte  velocity  acquired  in  falling  down  a  smooth 
inclined  plane  is  the  same  as  would  be  acquired  in  falling 
freely  through  the  perpendicular  height  of  the  plane. 


tCLg. 

is  incladed  in 
rticles  moving 
ill  be  made  to 
if  tbe  {Hirticle 
>m(l),(a),(3) 


(2) 
(3) 

(4) 
;ording  as  the 

}  get  from  (4), 

(ft) 

(8) 

(7) 
cfo  in  falling 

if  V  be  the 


le  same  at  P 
kl  space  OC  ; 
V«  a  stnoolh 
in  falling 
\lan0. 


DBaCSNT  DOWN  CBOBDS  OF  A  CtBCLX. 

CJOB.  2. — When  the  particle  is  projected  up  the  plane  teith 
a  given  veloef^y,  to  find  how  high  it  will  ascend,  and  the  time 
ofascetU. 

From  (4)  we  have 

1^  =  w,'  —  2^  sin  a'S. 

When  V  =  0  the  particle  will  stop ;  henoe,  the  distance  it 
will  ascend  will  be  given  by  thb  equation 

0  =  r,»  —  Zgaina'S, 


8  = 


2g  sin  « 
To  find  the  time  we  have  from  (2) 

V  =  v^  —g  aina-t; 
and  the  particle  stops  when  v  =  0,  in  which  case  we  have 

i  = 


g  ana 


From  (6)  we  derive  the  following  canons  and  nseful 
result 


145.  The  Times  of  Descent  down 
all  Chords  drawn  from  the  Highest 
Point  of  a  Vertical  Circle  are  eqnaL— 

Let  AB  be  the  vertical  diameter  of  the 
circle,  AG  any  cord  through  A,  a  its 
inclination  to  the  horizon  ;  join  BO  ;  then 
if  ^  be  the  time  of  descent  down  AG  we 
have  from  (6)  of  Art.  144 


Fi»7S 


Bat 


AC  =  yP  sin  d. 

AC  'js  AB  sin  u ; 


aMiWNtaMliUHWUM'  ^mmmmmimiammttm 


iiUfM  OF  qmoKMar  onacavt. 


.'.    AB»5fcf#», 


/2AB 


which  is  constant,  and  shows  that  the  time  of  falling  down 
any  chord  is  the  same  as  the  time  of  falling  down  the 
diameter. 

Cor. — Similarly  it  may  be  shown  that  the  times  of 
descent  down  all  chords  drawn  to  B,  the  lowest  point, 
are  equal ;  that  is,  the  time  down  OB  is  eqaal  to  that 
down  AB. 

146.  The  Straight  Idna  of  QoickoBt  Descent  from 
(1)  a  Qiven  Point  to  a  Qiven  Straight  lone  (2)  from 
a  Given  Point  to  a  Given  Onrve. 

(1)  Let  A  be  the  given  point  and  BO  c 
the  given  Uqq.  Through  A  draw  the 
horizontal  line  AO,  meeting  OB  in  0; 
bisect  the  angle  ACB  by  00  which  inter- 
sects in  0  the  vortical  line  drawn  through 
A ;  from  0  draw  OP  perpendicular  to  BO; 
Join  AP ;  AP  is  the  required  line  of  quick- 
est descent 

For  OP  is  evidently  equal  to  OA,  and  therefore  the 
circle  described  with  0  as  centre  and  with  OP  (=  OA)  for 
radius,  will  touch  the  line  BO  at  P,  and  since  the  time  of 
falling  down  all  chords  of  this  circle  from  A  is  the  same, 
AP  must  be  the  line  of  quickest  descent. 

(2)  To  find  the  straight  line  of  quickest  descent  to  a 
given  ewrvt,  all  that  is  required  is  to  draw  a  circle  having 
the  given  point  as  the  upper  extremity  of  its  vertical 
diameter,  and  tangent  to  the  curve.  Hence  if  DE  (Fig. 
76)  be  the  curve,  A  the  point,  draw  AH  vertical ;  and,  with 
centre  in  AH,  describe  a  circle  p^wsing  through  A,  and 


IHI 


m 


1  falling  down 
ing  down  the 

the  times  of 

I  lowest  point, 

equal  to  that 


Descent  from 
U»9  (2)  from 


n«.7«  ^ 


therefore  the 
}P  {-  OA)  for 
I  the  time  of 
ia  tiie  same, 

descent  to  a 
circle  having 
jf  its  Tertical 
if  DE  (Fig. 
Ell ;  and,  with 
>agh  A,  «ad 


XXAMPLSa. 


249 


touching  DE  at  P,  then  AP  is  the  required  line.  For,  if  we 
take  any  other  point,  Q,  in  DE,  and  draw  AQ  cutting  the 
circle  in  q,  then  the  time  down  AP  =  time  down  A.q<i 
time  down  AQ.    Hence  AP  is  the  line  of  quickesl  descent. 

Thu  problem  of  6ndiDg  the  line  of  quickest  descent  from  a  point  to 
a  line  or  curve  is  thus  found  to  resolve  itself  into  the  purely  geometric 
problem  of  drawing  a  circle,  the  highest  point  of  which  shall  be  the 
given  point  and  which  shall  touch  the  given  line  or  curva 


EXAMPLES.* 

1.  If  the  earth  travels  in  its  orbit  600  million  miles  in 
365jt  days,  with  uniform  motion,  what  is  its  velocity  in 
miles  per  second  ?  Ana.  19 -01  miles. 

2.  A  train  of  cars  moving  with  a  velocity  of  20  miles  an 
hour,  had  been  gone  3  hours  when  a  locomotive  was 
dispatched  in  purstiit,  with  a  velocity  of  25  miles  an  hour; 
in  what  time  did  the  latter  overtake  the  former  ? 

Ana.  12  hours. 

3.  A  body  moving  from  rest  with  a  uniform  acceleration 
describes  90  ft.  in  the  5th  second  of  its  motion  ;  find  the 
acceleration,/,  and  velocity,  v,  after  10  seconds. 

Ans.  /  =  20 ;  v  =  200. 

4.  Find  the  velocity  of  a  particle  which,  moving  with  an 
acceleration  of  20  ft.  per  sec.  has  traversed  1000  ft. 

Ans.  200  ft.  per  sec. 

5.  A  body  is  observed  to  move  over  45  ft.  and  55  ft.  in 
two  eaccessive  seconds ;  find  the  space  it  would  describe  in 
the  20th  second.  Ana.  195  ft. 

6.  The  velocity  of  a  body  increases  every  hour  at  the  rate 
of  360  yards  per  hour.  What  is  the  acceleration,  /,  in  feet 
per  second,  and  what  is  the  space,  a,  described  from  rest  ? 

Ans.  f=  0-3;  v  =  60  ft. 

*  In  the«c  examplci  Uke  g--9an. 


MM 


WO 


MXAMPLKM. 


7.  A  body  is  moting,  at  a  given  instant,  at  the  rate  ^ 
8  ft  per  sec.;  at  the  end  of  5  aecs.  its  >tl(K!ity  is  19  ft.  per 
sec.  Aeeuming  its  acceleration  to  be  uniform,  what  was  its 
velocity  at  the  end  of  4  sees.,  and  what  will  be  its  velocity 
at  the  end  of  10  sees.  ?  Ana.  16-  8 ;  30. 

8.  A  body  is  moving  at  a  given  instant  with  a  velocity  of 
30  miles  an  hoar,  and  comes  to  rest  in  II  sees.;  if  the 
retardation  is  uniform  what  was  its  velocity  5  sees,  before  it 
stopped  ?  Ans.  20  ft.  per  sec. 

9.  A  body  moves  at  the  rate  of  12  fi  a  sec  with  a 
nniform  acceleration  of  4;  (1)  state  exactly  what  is  meant 

the  number  4  ;  (2)  suppose  the  aci deration  to  go  on  for 
i  6  .sees.,  and  then  to  cease,  what  distance  will   the  body 

1;  describe  between  the  ends  of  the  6th  and  12th  sees.? 

Ans.  224  ft. 

10.  A  body,  whose  velocity  undergoes  a  uniform  retarda- 
tion of  8,  describes  in  2  sees,  a  distance  of  30  ft.;  (1)  what 
was  its  initial  velocity  ?  (2)  How  much  longer  than  the 
2  sees,  would  it  move  before  coming  to  rest  ? 

Am.  (1)23;  (2)  |  sec. 

11.  A  body  whose  motion  is  uniformly  retarded,  changes 
its  velocity  from  24  to  6  while  describing  a  distance  of  12 
ft.;  in  what  time  does  it  describe  the  12  ft.? 

Ans.  0-8  sec. 

12.  The  velocity  of  a  body,  which  is  at  first  6  ft.  a  sec, 
nndergoos  a  uniform  acceleration  of  3 ;  at  the  end  of  4  sec& 
the  acceleration  ceases  ;  how  far  does  the  body  move  in  10 
sees,  from  the  beginning  of  the  motion  ?        Ans.  156  ft 

13.  A  body  moves  for  a  quarter  of  an  hour  with  a  uni- 
form acceleration ;  in  the  first  5  minutes  it  describes  350 
yards;  in  the  second  5  minutes  420  yards;  what  is  the 
whole  distance  described  in  a  quarter  of  an  hour  ? 

Ans.  1260  yds. 


»  at  the  rate  of 
3ity  is  19  ft.  per 
m,  what  was  its 

I  be  its  velocity 
w.  16-8;  30. 

ith  a  velocity  of 

I I  sees.;  if  the 
5  sees,  before  it 
20  ft.  per  sec. 

^  a  sea  with  a 
what  is  meaut 

ion  to  go  on  for 
will  tho  body 

ith  sees.? 
Ans.  284  ft. 

oiform  retarda- 
50  ft.;  (1)  what 
onger  than  the 

S3;  (2)  I  sec. 

ardcd,  changes 
distuuce  of  12 

ms.  0-8  sec. 

frst  6  ft.  a  sec, 
end  of  4  sees; 
move  in  10 
ins.  156  ft 

|ur  with  a  nni- 

describes  3.50 

what  is  the 


jur  i 


1260  Yds. 


MXAMPLSS. 


Ml 


14.  Two  sees,  after  a  body  is  let  fall  another  body  is 
projected  yertioally  downwards  with  a  velocity  of  100  ft 
per  sec. ;  when  will  it  overtake  the  former  i* 

Ana.  1|  sees. 

15.  A  body  is  projected  upwards  with  a  velocity  of  100 
ft  per  sec.;  find  the  whole  time  of  flight     Ans.  li^  sees. 

IG.  A  balloon  is  rising  uniformly  with  a  velocity  of  10  ft. 
per  sec,  when  a  man  drops  from  it  a  stone  which  reaches 
the  ground  in  3  sees.;  find  tho  height  of  the  balloon,  (1) 
when  the  stone  was  dropped ;  and  (2)  when  it  reached  the 
ground.  Ans.  (1)  114  ft;  (2)  144  ft 

17.  A  man  is  standing  on  a  platform  which  descends 
with  a  uniform  acceleration  of  6  ft  per  sec. ;  after  having 
descended  for  2  sees,  he  drops  a  bull ;  what  will  be  the 
velocity  of  the  ball  after  2  more  seconds  ?        Ans.  74  ft. 

18.  A  balloon  has  been  ascending  vertically  at  a  uniform 
rate  for  4.6  sees.,  and  a  stone  let  fall  from  it  reaches  the 
ground  in  7  sees.;  find  the  velocity,  v,  of  the  balloon  and 
the  height,  «,  fh)m  which  the  stone  is  let  fall. 

Ans.  V  =  174|  ft  per  sec.;  s  =  784  ft  If  the  balloon 
is  still  ascending  when  the  stone  is  let  fall  v  =  68-17  ft. 
per  sec;  s  =  306-76  ft? 

19.  With  what  velocity  must  a  particle  bo  projected 
downwards,  that  it  may  in  t  sees,  overtake  another  particle 
which  has  already  fallen  through  a  f t  ? 


Ans.  V  -.  T  +  VZc^. 


20.  A  person  while  ascending  in  a  balloon  with  a  vertical 
velocity  of  V  ft  per  sec,  lets  fall  a  stone  when  he  is  k  ft. 
above  the  ground ;  required  the  time  in  which  the  ntony 
will  reach  the  ground.  ^        y  +  ^yt  —  2gh 


Ans. 


20S 


XXAMPLS& 


21.  A  body,  A,  is  projected  vertically  downwards  from 
the  top  of  a  tower  with  the  velocity  V,  and  one  sec.  after- 
wards another  body,  B,  is  let  fall  from  a  window  a  ft.  from 
the  top  of  the  tower  ;  in  what  time,  t,  will  A  overtake  B  ? 

2a  +  g 


Ana.  t  = 


^(y  +  g) 


32.  A  stone  let  fall  into  a  well,  is  heard  to  strike  the 
bottom  in  t  seconds  ;  required  the  depth  of  the  well,  sup- 
posing the  velocity  of  sound  to  be  a  ft.  per  sec. 

^ a_~|» 

'iff       V¥gJ ' 


Ans.     \/  at 


+ 


23.  A  stone  is  dropped  into  a  well,  and  after  3  sees,  the 
sound  of  the  splash  is  heard.  Find  the  depth  to  the 
surface  of  the  water,  the  velocity  of  sound  being  1127  ft. 
per  sec. 

24.  A  body  is  simultaneously  impressed  with  three 
uniform  velocities,  one  of  which  would  cause  it  to  move 
10  ft.  North  in  2  sees. ;  another  12  ft.  in  one  sec.  in  the 
same  direction  ;  and  a  third  21  ft.  South  in  3  sees.  Where 
will  the  body  be  in  5  sees.  ?  Am.  50  ft.  North. 

25.  A  boat  is  rowed  across  a  river  1^  miles  wide,  in  a 
directioi^  making  an  angle  of  87°  with  the  bank.  The 
bviat  travels  at  the  rate  of  5  miles  an  hour,  and  the  river 
nu\s  at  the  rate  of  2.3  miles  an  hour.  Find  at  what  point 
of  tlie  opposite  bank  the  boat  will  land,  if  the  angle  of  87° 
be  made  against  the  stream. 

Ans.  898  yards  down  the  stream  from  the  opposite 
point. 

26.  A  body  moves  with  a  velocity  of  10  ft.  yter  sec.  in  a 
given  direction  ;  find  the  velocity  in  a  direction  inclined  at 
an  angle  of  30°  to  the  original  direction. 

Am.  6  V3  ft.  per  sec. 


i(^^!^p»i^S!W««!^ta^'''i 


EXAMPLES. 


963 


wrnwards  from 
one  sec.  af  ter- 
dow  a  ft.  from 
overtake  B  ? 

1  to  strike  the 
the  well,  aup- 
c. 


ter  3  sees,  the 
depth  to  the 
being  1127  ft. 


I    with    three 

se  it  to  move 

le  sec.  in  the 

sees.    Where 

ft.  North. 

C8  wide,  in  a 

)ank.     The 

ind  the  river 

what  point 

angle  of  87° 

the  opposite 


t)er  sec.  m  a 
u  inclined  at 

rt.  per  sec. 


27.  A  smooth  piano  is  inclined  at  an  angle  of  30°  to  the 
horizon  ;  a  body  is  started  up  the  plane  with  the  velocity 
Hg;  find  when  it  is  distant  9^  from  the  starting  point 

Arts.  2,  or  18  sees. 

28.  The  angle  of  a  plane  is  30° ;  find  the  velocity  with 
which  a  body  mnst  be  projected  up  it  to  reach  the  top, 
the  length  of  the  plane  being  20  ft. 

Ana.  8  VlO  ft.  per  sec. 

29.  A  body  is  projected  down  a  plane,  the  inclination  of 
which  is  45°,  with  a  velocity  of  10  ft.;  find  the  space 
described  in  2^  sees.  Ant.  95.7  ft.  nearly. 

30.  A  steam-engine  atarts  on  a  downward  incline  of 
1  in  200*  with  a  velocity  of  7|^  miles  an  hour  neglecting 
friction ;  find  the  space  traversed  in  two  minutes. 

Ana.  824  yards. 

31.  A  body  projected  up  an  incline  of  1  in  100  with  a 
velocity  of  15  miles  an  hour  just  reaches  the  summit ;  find 
the  time  occupied.  Ana.  68.75  sees. 

32.  From  a  point  in  an  inclined  plane  a  body  is  made  to 
slide  up  the  plane  with  a  velocity  of  16.1  ft.  per  sec.  (1) 
How  far  will  it  go  before  it  comes  to  rest,  the  inclination 
of  the  plane  to  the  horizon  being  30°  ?  (2)  Also  how  far 
will  the  body  be  from  the  starting  point  after  5  sees,  from 
the  beginning  of  motion  ? 

Ana.  (1)  8.05  ft. ;  (2)  120.75  ft.  lower  down. 

33.  The  inclination  of  a  plane  is  3  vertical  to  4  hori- 
zontal ;  a  body  is  made  to  slide  up  the  incline  with  an 
initial  velocity  of  36  ft.  a  sec. ;  (1)  how  far  will  it  go  before 
beginning  to  return,  and  (2)  after  how  many  seconds  will 
it  return  to  its  starting  point? 

Ana.  (l)33f  ft.;  (2)  3|  sees. , 


*  An  Incline  6r  1  In  SOO  mMns  here  1  fnot  vcrttcally  to  •  length  of  iOO  ft.,  Uumgh 
U  to  nMd  b;  Bnglneen  to  mean  1  foot  vertically  to  nO  ft.  AoH«mta^. 


MMHMiMi 


Ui 


MXAMPLM$, 


34.  There  is  so  inclined  plane  of  6  vertical  to  12  hori- 
zontal, a  body  «lidea  down  52  ft  of  its  length,  and  tLo 
passes  without  b«  of  velocity  on  to  the  horizoittl  pluuT 
after  how  long  from  the  beginning  of  the  motion  will  it  hj 
at  a  distauce  of  100  f  L  from  the  foot  of  the  incline  ? 

Ans.  6.7  sees. 

i/iu^  I^^  ''  projected  up  an  inclined  plane,  whose 
length  IS  10  times  its  height,  with  a  velocity  of  si  tiZ 
«ec. ;  m  what  time  will  its  velocity  be  destroyed  ?  ' 

Am.  9|  sees.,  if  ^r  =  32. 
30.  A  body  falls  from  rest  down  a  given  inclined  plane; 
compare  the  tiwes  of  describing  the  first  and  last  halves 

-4ns,  As  1  :  ^2  -f  1. 

37.  Two  bodies,  projected  down  two  pianos  in-^lmed  to 
he  honzon  at  anglee  of  45»  and  60°.  describe  in  the  same 

time  sp-oces  re8{.ectively  as  ^/i  :  ^3;  find  the  ratio  of  the 
initial  vokKjities  of  the  projected  bodies. 

Ana.  V2  :  1/3. 

38.  Thmugh  what  chord  of  a  circle  must  a  body  fall  to 

Zeter^  '''  "'"'^-^  '''"''   ''  '*"*"^  "^-V  th^ 
Am.  The  chord  which  is  inclined  at  60'  to  the  vertical 

il\  ^^^  ^^^  ""'^."^''^^  ""'^^  ^^'"'^  *  ^y  «^««5<1  be  pro. 
jocted  down  an  inclined  plane,  /.  «o  that  the  time  of 

running  down  the  plane  shall  be  equal  to  the  time  of 
falling  down  the  height,  h. 

.  A      *  «in  «\ 

nf  *r;.  ^'f!''^^  inclination  of  this  plane,  when  a  velocity 
of  fths  that  duo  to  the  height  is  sufficient  to  render  the 
timoH  of  running  down  the  piano,  and  of  falling  down  the 
height,  equal  to  eacli  othor.  ^„,,  30°. 


tical  to  12  hori- 
eugth,  and  tlieu 
lorizoQtal  piano ; 
aotion  will  it  be 
incline? 
Ans.  6.7  sees. 

d  plaue,   whose 

ity  of  30  ft.  per 

iyed? 

38.,  if  ^  ~  33. 

inclined  plane; 
and  last  halves 

li  Vi  -h  1. 

nes  in'ilined  to 
be  in  the  eanio 
:he  ratio  of  the 

f.  V2  :  Vs. 

a  body  fall  to 
through  the 

the  rerticaL 

ihould  be  pro- 
the  tiiuo  of 
the  time  of 


o 


h  Bin  n\ 

leu  a  velooitj 
to  render  the 
ng  down  the 
Ans.  80°. 


SXAMPLJB8. 


41.  Through  what  chord  of  a  circle,  drawn  from  tlie 
bottom  of  the  yertical  diameter  mu^t  a  body  descend;  so  sm 

to  acquire  a  velocity  equal  to   -th  part  of  the  velocity 

acquired  in  falling  down  the  vertical  diameter  ? 
A»s.  If  0  denoi'  aie  angle  between  tho  required  chord 

and  the  vertical  diameter  cos  0  =  -  • 


42.  Find  the  incUnation,  d,  of  the  radius  of  a  circle  to 
the  vertical,  such  that  a  body  running  down  will  describe 
the  radius  in  the  same  time  that  another  bodji  n>quir&a  to 
full  down  the  vertical  diameter.  Ans.  6  =  60°. 

43.  Find  the  inclination,  0,  to  the  vertical  of  tho  diam- 
eter down  which  a  body  falling  will  describti  the  last  half 
in  the  same  time  as  the  vertical  diamekvr. 

8a/|j-4 

/8   "" 


Ah$.  co«  6  = 

44.  Show  that  the  times  of  descent  down  all  the  radii  of 
curvature  of  tho  cycloid  (Fig.  40,  Calculus)  ore  equal;  that 

8r 


is,  the  time  down  PQ  is  equal  to  the  time  down  O'A  = 


y 


46.  Find  the  Inclination,  ^,  to  the  horizon  of  an  inclined 
plane,  so  that  the  time  of  descent  of  a  particle  down  the 
length  may  be  »  tiroes  that  down  tho  height  of  tho  plane. 


Ans.  0  =  sin 


.1 


46.  Find  the  line  of  quickest  descent  from  tho  focus  to 
a  parabola  whose  axis  is  verticul  and  vertex  upwards,  and 
show  that  its  length  is  equal  to  that  of  the  latus  rectum. 

47.  Find  the  lino  of  quickest  descent  from  the  foous  of  a 
parabola  to  the  curve  when  the  axis  is  horisMirital. 


"IMII 


256 


SXAMPLSa, 


48.  Find  geometrically  the  line  of  quickest  descent  (1) 
tfom  a  point  within  a  circle  to  the  circle  :  (2)  from  t\  circle 
to  a  point  without  it. 

49.  Find  gpometrically  the  straight  line  of  longest 
descent  from  a  circle  to  a  point  without  it,  and  which 
lies  below  the  circle. 

60.  A  man  six  feet  high  walks  in  a  straight  line  at  the 
rate  of  four  miles  an  hour  away  from  a  street  lump,  the 
height  of  which  lis  10  foot :  eupposing  the  man  to  starh 
from  the  L*mp-i)ost,  finu  the  rate  at  which  the  end  of  his 
shadow  travels,  and  also  the  rate  at  which  the  end  of  bis 
shadow  Be])arate8  from  himself. 

Ana.  ;j!  ,dow  travels  10  miles  an  hour,  and  gains  on 
himseii  (3  miles  an  hour. 

'•i.  Two  bodies  fall  in  the  same  time  from  two  given 
pointB  in  space  in  the  same  vertical  down  two  straight 
J-n. «  drawn  to  any  point  of  a  surface ;  show  that  the  sur- 
laco  is  an  equilateral  hyperboloid  of  revolution,  having  the 
given  jwintfl  as  vertices. 

62.  Find  the  form  of  a  curve  in  a  vertical  plane,  such 
that  if  ht'iivy  pa/tioles  be  simultaneously  let  full  from  ca^h 
point  Mt  it  80  as  to  nlide  freoly  along  the  normal  at  that 
point,  they  may  all  reach  a  given  horizontal  straight  Hue  at 
the  same  instant. 

53.  Show  that  the  time  of  quickest  descent  down  a  focal 
chord  of  a  parabola  whoso  axis  is  vertical  is 


m 


where  I  is  the  latus  rectum. 

54.  Particles  slide  from  rest  at  the  highest  point  of  a 
vertical  circle  down  chords,  and  ore  then  allowed  to  moye 


«<'■«' 


t  deacent  (1) 
from  n  circle 


i  of  longest 
t,  and  which 

it  line  at  the 
•eet  lump,  the 
man  to  Btarti 
le  end  of  his 
le  end  of  bis 

md  gains  on 

om  two  given 
two  straight 
lit  the  sur- 
huving  the 

!  I;ine,  such 

1  from  each 

rmul  at  that 

'uighk  Hue  at 

lown  a  focal 


fioint  of  a 
to  move 


EXAMPLES. 

freely  ;  show  that  the  locns  of  the  foci  of  their  paths  is  a 
circle  of  half  the  radius,  and  that  all  the  paths  bisect  the 
vertical  radius. 

55.  If  the  particles  slide  down  chords  to  the  lowest  point, 
and  be  then  suffered  to  move  freely,  the  locus  of  the  foci  is 
a  cardioid. 

66.  Particles  fall  down  diameters  of  a  vertical  circle  ;  the 
locus  of  the  foci  of  their  subsequent  paths  is  the  circle. 


ii: 


CHAPTER    II. 

CURVILINEAR    MOTION. 

147  Remarlw  on  CnrTillnear  Motion.— The  mo- 
tion, vrliioh  waa  coneidered  in  the  laat  chapter,  wa8  that  of 
a  partiola  describing  a  rectilinear  path.  In  this  chapter  V.\e 
circnmstances  of  motion  in  which  the  path  is  ctirvilinear 
will  he  considered.  The  conception  and  the  definition  of 
Telocity  and  of  acceleration  which  were  given  in  Artfl.  134, 
135,  are  evidently  ad  applicable  to  a  particle  describing  a 
curvilinear  v"  ha  to  one  moving  along  a  straight  line; 
and  coDsequ  i.  Jie  fo>-mulaB  for  velocity  in  Arts.  143, 143, 
are  ap{>licablc  eiiher  to  rectilinear  or  to  curvilinear  motion. 
In  the  last  chapter  the  effects  of  the  corapoeition  and  the 
resolutiou  of  velocities  were  considered,  when  the  path 
taken  by  the  particle  in  consequence  of  them  was  stmight ; 
we  have  now  to  investigate  the  effects  of  velocities  and  of 
accelerations  in  a  more  general  way. 

148.  Composition  cf  Unifonn  Velocit7  and  Ac- 

o«ler«tlon.—SupiwBO  a  body  tends  to  move  in  one  direc- 
tion with  a  uniform  velocity  which  would  carry  it  from  A 
to  B  in  one  second,  and  also  suV>ject  to  an 
acceleration  thai  would  carry  it  from  A 
to  0  in  one  second ;  then  at  the  end  of 
the  second  the  body  will  bo  at  D,  the 
opposite  end  of  the  diagonal  of  the  par- 
allelogram ABDC,  just  as  if  it  had  moved 
from  A  to  B  and  then  from  B  to  D  in  the  second,  bnt  the 
body  will  move  in  th^  nivve  and  not  along  the  diagonal. 
For,  the  body  in  its  motion  is  making  progress  uniformly 
iu  the  direction  AB,  at  the  same  rate  iw  if  it  had  no  other 
motion;  and  at  the  Raniw  time  it  is  being  accelenUed  in  the 


OOMPOBtTION  OF  AOCMLMMATIOKa, 


N. 

ion. — The  mo- 
■jN,  waH  that  of 
this  chapter  the 
h  is  curvilinear 
he  definition  of 
m  in  Artfl.  134, 
ilo  describing  a 
I  straight  line ; 
J  Arts.  142, 143, 
rilinear  motion. 
»itiou  and  the 
rhen   the  path 

was  sti-aight ; 

ocitit'B  and  of 

ity  and  Ac- 

in  one  direc- 
rry  it  from  A 


cond,  bnt  the 

the  diagonal. 

■em  uniformly 

had  no  othor 

united  in  the 


9M 


direction  AC,  aa  fast  as  if  it  had  no  other  motion.  Hence 
the  body  will  reach  D  as  far  from  the  tine  AO  as  if  it  bad 
moved  over  AB,  and  as  far  fi'om  AB  as  if  it  had  moved 
over  AC ;  but  since  the  velocity  along  AC  is  not  uniform, 
the  spaces  described  in  equal  intervals  of  times  will  not  be 
equal  along  AC  while  they  are  equal  along  AB,  and  there- 
fore the  points  a^,  a,,  o,,  will  not  be  in  a  straight  line.  lu 
this  case,  therefore,  the  path  is  a  curve. 

149.  Composition  and  Resolution  of  Accelera- 
tions.— If  a  body  i«  Bnbjeot  to  two  different  accelerations 
in  different  directions  the  sides  of  a  parallelogram  may  bo 
taken  to  represent  the  ComponeiU  Accelerations,  and 
the  diagonal  will  represent  the  Resultant  Acceleration, 
although  the  path  of  the  body  may  be  along  some  other 
line. 

Rem. — These  results  with  those  of  Arts.  142, 148,  may  be 
summed  up  in  one  general  law:  When  a  body  tmds  to 
move  with  several  different  velocities  ^'n  different  directions, 
the  body  will  be,  at  the  end  of  any  given  time,  at  the  same 
point,  as  if  it  had  moved  with  each  velocity  separately. 
This  is  the  fundamental  law  of  the  composition  of  veloci- 
ties, and  it  shows  that  all  problems  which  involve  tenden- 
cies to  motion  in  different  direotionH  simultaneously,  may 

be  treated  as  if  those  tendcDcios  were  successive.* 
(jPg 
If  -js  be  the  acceleration  along  the  curve,  and  («,  y,  t) 

be  the  place  of  the  moving  particle  at  the  time,  ij  it  is 
evident  that  th(>  component  accelerations  parallel  to  the 

llf*'  ^'  3^'     ^^^°**'''°»  ^^^^  ^y  "»  **"  ««i 


ares  are 


W8 


have 


df 


a*'* 


dP 


«»; 


di* 


—  «•; 


and  V<«(^  -I-  o/  +  «i^  i«  the  rumUani  acceleration. 


•  flwi  H^irairkii  oo  Newlon'pi  M  Ur,  Art.  I6E>. 


IKHBHI 


360 


coMPoamoif  or  AccELSRATwna. 


Also  if  a,  P,  y,  be  the  angles  which  the  direction  of 
motioQ  makes  with  the  axes,  we  have 

^  =  ^-,0080  =  0,; 


d»2 


The  acceleration        is  not  generally  the  complete  resultant  of  the 

threr  component  acceleratious,  bat  is  eo  only  when  the  path  ig  a 

9trai}i;'>^  line  or  the  velocity  ia  lero.    It  is,  however,  the  only  part 

(P* 
of  thew  iHwultant  which  has  any  effect  on  the  velocity,     -r-^  is  the 

sum  of  the  njaolved  partB  of  the  component  accelerations  in  the  direc- 
tion of  motion,  as  the  following  identical  equation  showa: 


which  follows  immediately  from  (1)  of  Art.  187  by  diil^rentlation. 
Accelerations  are  therefore  Bubject  to  the  same  laws  of  composition 
and  resoluti'm  as  velocities  ;  and  ronseqaently  the  acceleration  of  the 
particle  along  any  lino  is  the  sum  uf  the  resolved   |)art8  of  the  axial 

accelerations  along  that  line.  Thus  to  find  .^,  the  acceleration  along  •, 

^  ('  dj? 

\,^  has  to  be  multiplied  by  v-,  which  is  the  direction  cosine  of  the 

dC  '^  '  d$ 

small  arc  ds.    'Hie  other  part  of  the  resultant  is  at  right  biihUk  to 

this,  and  Its  only  effect  iii  to  change  the  dircetion  of  the  motion  of  the 

l>oint.    (S<><>  Tait  and  Steele's  Dynamics  of  a  Partlcl.',  also  Thomson 

and  Tait's  Nat  Phil.) 

The  following  are  examples  in  wluch  the  preceding  ex- 
priHsio!  >  ar>'  applii-d  to  onaei  in  which  the  laws  of  velocity 
and  of  ac<^ler»tion  ore  given. 


ONB. 

the  direction  of 


9  resultant  of  the 

ten  the  path  ia  a 
fer,  the  only  part 

locSty.     ^  ia  the 

tiona  in  the  direc 
tiowa: 


y  diflerenttation. 
of  composition 
cel«ratiun  of  the 
•arts  of  the  axial 

eleration  along  », 

on  coalne  of  the 

right  ati(^li«  to 
le  motion  of  the 
t),  also  Thomaon 


)roceding  cx- 
vrs  of  velocity 


EXAMPLES. 


1.  A  particle  moves  so  that  the  axial  compononts  of  its 
velocity  vary  as  the  corres  wnding  co-ordinates ;  it  is 
required  to  find  the  equation  of  its  path ;  and  the  accel- 
erations along  the  axes. 


Here 


dx 
di 


.—  kx 


dy 
dt 


=  h> 


.'.    ^  =  ^  =  kdt; 

'      y 

if  (a,  b)  is  the  initial  »faice  of  tb«  particle, 

is  the  equation  of  the  path. 
And  the  axial  accelerations  are 


H-m 


di» 


dfl 


2.  A  wheel  rolls  along  a  straight  line  with  a  nniform 
velocity  ;  compare  the  velocity  of  a  given  point  in  the  ilr- 
cumference  with  that  of  the  centre  of  the  wheel. 

Let  the  line  along  which  the  wheel  rolls  be  the  axis  of  x, 
and  let  v  be  the  velocity  of  its  centre;  then  a  point  in  its 
circumference  describes  a  cycloid,  of  which,  the  origin 
being  taken  at  its  starting  {mint,  the  equation  is 


a  —  a  vera 


a 


(2ay  -  y)*; 


iiiiii 


It 


!i- 


i  ,1 


EXAMPLMa. 

^  _  d^  __  _     <fe 

y*       (3a  -"y)*  ~  (2a)** 


Bat 


'^^  «'       (iJay-S'*)*    * 

,      da  _ds    dy       /2y\i 

which  is  the  velocity  of  the  point  in  the  circumference  of 
the  wheel.  Thus  the  velocity  of  the  highest  point  of  the 
wheel  is  twice  as  great  as  that  of  the  centre,  while  the 
point  that  is  in  contact  with  the  straight  line  has  no 
velocity.    (See  Price's  Aaal.  Mech's.,  Vol,  I,  p.  416.) 

3-  ^^  ^  —  ^'y>  ^  —  *•»>  show  that  the  path  is  an  equi- 
lateral hyperbola  and  that  the  axial  components  arc 

4.  A  particle  describes  an  eDiiwe  so  that  the  a^component 
of  its  velocity  is  a  constant,  a  ;  find  the  y-com|)onent  of  \U 
vt'l(Kjity  and  acceleration,  and  the  time  of  describing  the 
ellipse. 

Ijet  the  fif/iia«i/(n  of  the  ellipse  be 


^i 


\\ 


and  lot  {x,  y)  bo  the  position  of  the  parHfiht  at  the  time  /  j 
then  ^--.    "-'    ^V  ^ 


dx  .    dy  hh; 

W^"'    and    J=-;^; 


^  ^dy   (h  _ 
dt       dx'  dt  ~ 


«»    y' 

which  is  the  y-component  of  the  velocity. 


dy 


jircumference  of 
lest  point  of  the 
entre,  while  the 
?ht  line  has  no 
[,p.416.) 

path  is  ao  oqui- 
lenta  are 


bo  ^-Component 

)m])onent  of  itti 

doBcribiug  the 


at  the  timo  t 


wmmm 


mmmm^. 


MXAMPLB. 


363 


Also 


9y  _ 

dP  ~ 


dx         fly 


hence  the  acceleration  parallel  to  the  axis  of  y  varies 
inversel;  as  the  cube  of  the  ordinate  of  the  ellipse,  and  acts 
towards  the  axis  of  x,  as  is  shown  by  the  negative  sign. 

The  time  of  passing  from  the  extremity  of  the  minor 
axis  to  that  of  the  major  axis  i&  fonnd  by  dividing  a  by  a, 
the  constant  velocity  parallel  to  the  axis  of  x,  giving 

,  and  the  time  of  describing  the  whole  ellipse  is  —  • 
If  the  orbit  is  a  circle  b  =  a,  and  the  acceleration  par- 

allol  to  the  axis  of  w  is 3— 

If  the  v'olooity  parallel  to  the  ^-axis  is  constant  and  equal 
to  P,  then 

di *»  •  V ' 


dh: 


and  the  periodica  time  s= 


4ft 


a? 


1  ;  find 


5.  A  particle  descrihes  the  hyperbola  -,  —  ^ 

(1)  the  acceleration  parallel  to  the  aiis  of  a;  if  the  velocity 
parallel  to  the  axis  of  ^  is  a  constant,  0,  and  (3)  find  the 
acceleration  parallel  to  the  y-axis  if  the  velocity  parallel  to 
the  X-axis  is  a  constuut  a. 
(1)  Here  we  have 


di  -^' 


and 


<ii 


^  =  -  .  -• 


ii!(iii««>t>m<V'iVtiiiiMii?i«« Mmifmimmn%wniSt>'>mi-iriysr,«isim>fMi''  •■  iUM 


u 


264 


EXAMFhBS. 


dx 
dt 


dx    dy 
dy    ^ 


/3rt»   y 


which  is  the  velocity  parallel  to  the  a;-axi& 


Also 


%         dx 


_  ^ 


hence  the  acceleration  parallel  to  the  ar-axis  varies  inversely 
as  the  cube  of  the  abscissa,  and  *he  »-component  of  the 
velocity  is  increasing. 

(2)  Here  we  have  , 


dx 
dt 

=  «; 

dy_ 

dt  ~ 

Z 

dh,  _ 

dt»  ~ 

:   — 

ay 

and 


hence  the  acceleration  parallel  to  the  y-axis  is  negative  and 
the  y-component  of  the  velocity  is  decreasing. 

6.  A  particle  describes  the  parabola,  «»  +  y*  =  a*,  with 
a  constant  velocity,  c ;  find  the  accelerations  parallel  to  the 
axes  of  X  and  y. 


Here  we  have 


da 


and 


d'X 


dt 
■A- 


=  c; 


ds 


y 


*       {x  +  y)^* 


m 


varies  inversely 
mponent  of  the 


is  negative  and 
'g- 

-  y^  =  a*,  with 
B  parallel  to  the 


'^MM 


m0mmw 


EXAMPLES. 


.♦. 

X 

X  +  y       X 

i*X 

y' 

and 

d^ 

~  dfi ' 

y    _ 

X  -\-  y  ~^  X 

+  y' 

differentiating 

we  get 

d^ 

2{x  +  yr 

^  _ 

d^  ~ 

(?{ax)^ 
■  2  (a;  +  vY 

865 


7.  A  particle  describes  a  parabola  with  such  a  varying 
velocity  that  its  projection  on  a  line  peipendicular  to  the 
axis  is  a  constant,  v.  Find  the  velocity  and  the  accelera- 
tion parallel  to  the  axis. 

Let  the  ec^uation  of  the  parabola  be 


then 


and 


y»  =  %px; 

dy 
~dt 


=  v, 


dx  _  dx   dy 
dt  ^  dy   dt 


vy. 


which  is  the  velocity  parallel  to  x 
Also 


dfi 


P 


which  shows  that  the  particle  is  moving  away  from  the 
tangent  to  the  curve  at  the  vertex  with  a  constant  accelera- 
tion. 


,YS' 


>-«'**«*«>t*tti«iM«iife^a 


>.'^..\ 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


^ 


^ 


#/. 


o 


A 


[/ 


^^ 


^ 


fA^ 


1.0 


I.I 


1.25 


•>5 
2.2 


IM 


iia. 


U.  11.6 


Hiotographic 

Sciences 
Corporation 


33  WIST  fciAIN  STRUT 

WnSTiR.N.Y.  143*0 

(7*6)  &>a-4S09 


■•  iwuij  mw— 


CiHM/ICMH 

Microfiche 

Series. 


CIHIVI/ICIVIH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions  /  Institut  Canadian  da  microraproductions  hiatoriquaa 


i 


*■•*'  ,.?a#t#"*^'  ■••■■   !^a?f**** 


PROJKCTIhM  IN  VACUO. 

Hence  ae  the  earth  acts  on  partiolee  near  its  surface  with 
a  constant  acceleration  in  vertical  lines,  if  a  particle  is 
projected  with  a  velocity,  r,  in  a  hoiiiontal  line  it  wU  move 
in  a  parabolic  path. 

160.  Motiun  of  Protjeetiles  in  Vacua— If  a  particle 
be  projected  in  a  direction  obliqne  to  the  bozixon  it  i« 
called  a  Projectile,  and  the  path  which  it  describes  is  called 
it«  Trajectory.  The  ca»e  which  we  ghali  here  considt-r  is 
that  of  a  particle  moving  in  vacuo  under  the  action  of 
gravity;  so  that  the  problem  is  that  of  the  motion  of  a 
firojeciile  in  vacuo;  and  hence,  as  gravity  does  not  »ii\ct 
its  horizontal  velocity,  it  resoivea  itself  into  the  purely 
kinematic  problem  of  t  particle  moving  so  that  its  hori- 
zontal acceleration  i?  0  and  its  vertical  acceleration  is  the 
constant,  g,  (Art  140). 

151.    TlM   Patji   of   • 

Projectile  in  Vacuo  is  a 
Parabola.— Let  the  plane 
in  which  the  particle  is  pro- 
jected be  the  piano  of  zy ; 
let  the  axis  of  x  bo  horizon- 
tal And  the  axis  of  y  vertical 
and  positive  upwards,  tb3 
origin  being  at  the  point  of 
projection;   let  the  velocity 

of  projection  =:  v,  and  let  the  line  of  projection  be  inclined 
at  an  angle  a  to  the  axis  of  x,  so  that  t;  co«  «,  and  v  bin  « 
are  the  rcbolved  parts  of  the  velocity  of  projection  along  the 
axes  of  r  and  y.  It  is  evident  that  the  particle  will  oon- 
tijue  to  move  in  the  plane  of  xy,  as  it  is  projected  in  i*, 
and  is  subject  to  no  toroo  which  would  tend  to  withdraw 
it  from  that  plane. 

Lot  {x,  y)  be  the  plaoe,  P,  of  the  partide  at  the  time  / ; 
then  the  equations  of  motioa  arj 


' 


its  snrface  with 
if  a  particle  is 
line  it  vnll  move 


O. — If  a  particle 
le  bomon  it  ic 
lescribes  is  called 
here  consider  is 
it  the  action  of 
the  motion  of  a 
'  does  not  aij:>>ct 
into  the  purely 
30  that  its  hon- 
ioeleration  is  the 


^ 


c 


3tion  be  inclined 
M  a,  and  V  bin  a 
lection  along  the 
)artiole  will  oon- 
1  projected  in  i*, 
tnd  to  withdraw 

e  at  tb0  time  / ; 


PMOJBOTILg  IN  VACUO.  867 

the  aeceleration  being  negative  since  the  y-component  of 
the  velocity  is  decreasing. 

The  first  and  second  integrals  of  these  equations  will 
then  be,  taking  the  limits  corresponding  to  <  =  /  uid 
<  =  0, 


_^„cos«;f  = 
g  5=  vQMoti  y 


vmaa  —  gt\  (1) 

vmuut--  yfl.  (it) 


Eqoationa  (1)  and  ^2)  give  the  coordinates  of  the  particle 
and  its  velocity  parallel  to  either  axis  at  any  time,  t. 
Eliminating  t  between  equations  (3)  we  obtain 


9) 


y  =  *to»«-2jfe- 


which  is  the  equation  of  the  traject<>ry,  and  shows  that  the 
particle  will  move  in  a  parabola. 

153.  TheParaawfeer;  th«  Range  JB;  tbaGhrMtaak 
Height  H;  Height  of  the  Directrix— Equation  (3)  of 
Art.  1^1  may  be  written 

.      2v*  sin  a  cos  a  2t/*  coi^  «e 

a^ _ x=s r !/, 


or 


ti*  sin  «  COS  o\'  2t)*  cos*  a  i 


9 


o\'  2t)*co8'a/        »"8ln»«\  .,. 


9 


^ 


By  comparing   this  with  the  equation  of  a  parabola 
referred  to  its  vertex  as  origin,  we  find  tor 

.,       .     ,         ...           .            V*  sin  a  cos  a  ,a\ 

the  absouM  of  the  ^  «)rtex  =  —— }        (2) 


ft' 


^^  I  -Jill! 


\ 


268  PROJSCTTLS  IN  VACUO. 

the  ordioate  of  the  vertex  =      "°  " ;  (3i 

the  parameter  (latus  rectum)  = ^^ — ^-       (4) 

And  by  transferring  the  origin  to  the  vertex  (1)  becomes 


.  2w»  co8»  a 

«•= :; — y 


(5) 


i 


which  is  the  equation  of  a  parabola  with  its  axis  vertical 
and  the  vertex  the  highest  point  of  the  curve. 

The  distance,  OB,  between  the  point  of  projection  and 
the  point  where  the  projectile  strilies  the  horizontal  plane 
is  called  the  Range  on  the  horizontal  •  plane,  and  is  the 
value  of  X  when  y  =  0.  Putting  y  =  0  in  (3)  of  Art.  161 
and  solving  for  x,  we  get 

tlui  horizontal  range  /2  =:  OB  =  ^  "°  ^";         (C) 

which  is  evident,  also,  geometrically,  as  OB  =  200 ;  that 
is,  the  range  is  eqiml  to  twice  the  abscissa  of  the  vertex. 

It  follows  from  (6)  that  the  range  is  the  greatest,  for  a 
given  velocity  of  projection,  when  a  =  45°,  in  which  case 

the  range  =  — • 

Also  it  appears  from  (6)  that  the  range  is  the  same  when 
o  is  replaced  by  its  complement ;  that  is,  for  the  same 
velocity  of  projection  the  range  is  the  same  for  two  differ- 
ent angles  that  are  complements  of  each  other.  If  «  =  46° 
the  two  angles  become  identical,  and  the  range  is  a 
maximum. 

OA  is  calltsd  the  greatett  height,  H,  of  the  projectile,  and 

ia  given  by  (3)  which,  when  a  =  45°  becomee  7-.  (7) 


f 


X 

I*  sin'  g 
''iff      ' 

2»»  cos»  g 


(3) 


(*) 


tex  (1)  becomoB 

(5) 

h  its  axis  vertical 
irve. 

of  projection  and 
e  horizontal  plane 
plane,  and  ia  the 
in  (3)  of  Art.  151 


w»  sin  2a 


(C) 


OB  =  200;  that 
of  the  vertex, 
the  greatest,  for  a 
5°,  in  which  case 


is  the  same  when 

is,  for  the  same 

ne  for  two  differ- 

(ther.    If  «  =  46° 

the  range   ia   a 


;he  projectile,  and 

4^'      (y 


Daes 


vsLOcirr  or  tbs  psojjbottlk. 
I%e  height  of  the  directrix  =  CD 

''iff' 


—  —^*Jt  A.  X  ^'^  COS* « 

-  2g       -^i     ~g 


269 


(8) 


Hence  when  «  =  45°  the  focns  of  the  parabola  lies  in 
tlie  horizontal  line  through  the  point  of  projection. 

153.  Tlie  Velocity  of  the  Partiole  at  any  Point  of 
its  Path. — Let  V  be  the  velocity  at  any  point  of  its  path, 

then       V^  =  (|)V  (I)*,  or  by  (1)  of  Ait  151 

=  w*  cos'  «  +  (v*  sin*  a  —  2v  sin  agt  +  g*P) 

s=v>  —  2gi/. 

To  acquire  this  veWity  in  falling  from  rest,  the  particle 

y» 
must  have  fallen  through  a  height  -^,  (6)  of  Art.  140,  or 

itsoqual  " 

~-ff  =  M8-MPhy(8) 

=  PS. 

Hence,  the  velocity  at  any  point,  P,  on  the  curve  is  that 
which  the  particle  would  acquire  in  falling  freely  in  vacuo 
down  the  vertical  height  8P;  that  is,  in  falling  from  (ho 
directrix  to  the  curve;  and  the  velocity  of  projection  at  O 
is  that  which  the  particle  would  acquire  in  falling  freely 
through  the  height  CD.  The  directrix  of  the  parabola  is 
therefore  determined  by  the  velocity  of  projection,  and  is  ai 
u  vortical  distance  above  the  point  of  projection  equal  to 
that  down  which  a  particle  flailing  would  have  the  velocity 
of  projection. 

154.  The  Time  of  Flight,  T,  along  a  Horizontal 
Plane.— Put  y  =  0  in  (a)  of  Art.  151,  and  solve  for  a-,  the 


i: ':. 


•mta  or  ruosr  op  pnojsoi'tMs. 
%ifi  sin  K  COB  CE 


870 

values  of  which  are  0  and 


9 


'    But  the  horizon- 
%v  sin  a 


tal  velocity  ia  v  cos  «.    Hence  the  time  of  flight  = 
which  varies  as  the  sine  of  the  inclination  tx)  the  axis  of  z. 


155.  To  FinA  the  Point  at  n^ch  a  PrOjoctlla  will 
Striko  a  Oiveii  IseliiMd  FbuM  paacing  tbrongh  the 
Point  of  Projeotion,  and  the  Time  of  Flight— Let  the 

inclined  plane  make  an  angle  /3  with  the  horizon ;  it  is 
evident  that  we  have  only  to  eliminate  y  between  y  =:  a:  tan 
/3  and  (3)  of  Art.  151,  which  gives  for  the  abscissa  of  the  '' 
point  where  the  projectile  meets  the  plane 

_  2t>'  cos  a  sin  («  —  jB) , 
^coe/3  * 

and  the  ordinate  i« 


x^  z= 


(1) 


_  a»»  COB  CT  tan  /3  sin  («  --  <3) 
'*  ~"  g  cos  /3 

Hence  the  time  of  flight 


T  = 


Xf      _  2v  sin  (g  —  /3) 
V  cos  «  g  cos  /3 


(2) 


156.  The  Direotlon  of  Protjeotloii  whiek  giTea  the 
Oreateat  Range  on  a  GUven  Plane. — The  range  on  the 
horizontal  plane  is 

t^  sin  2a 


which  for  a  given  value  of  v  is  greatest  when  «  =  7  {^^ 

152). 
The  range  on  the  inclined  plane  =  a^j  sec  0' 


—  8*^  COB  «  «'P  («  ""  <^). 


(1) 


•    But  the  horizon- 

f  flight  =  —^ 
D  \a  the  axis  of  x. 

a  Prdjdctlle  wlU 
icing  tisirongh  the 

ifriight— Letthe 
the  horizon ;  it  is 
between  y  —  x  tan 
'  the  abscissa  of  the 
le 


(1) 


"&) 


-J) 

8 


(2) 


k  ^hiek  giTtts  th« 

—The  range  on  the 


when  «  =  T  (Art. 


,  Bee  fi 


(1) 


ANOLX  or  MLBVATtOH  Of  PBOjaOTtLg. 


271 


To  find  the  value  of  a  which  makes  this  a  maximum,  we 
must  equate  to  zero  ita  derivative  with  respect  to  «,  which 
gives 

ooe  (2a  — /3)  =  0; 


and  henoe 


(9) 


(8) 


which  is  the  angle  which  the  direction  of  projection  makes 
wi*h  the  inclined  plane  when  the  range  is  a  maximum ; 
that  is,  the  projection  bisects  the  angle  between  the 
inclined  plane  and  the  vertical. 

In  this  case  by  subttitating  in  (1)  the  values  of  «  and 
(«  —  (3)  as  given  in  (2)  and  (3)  and  reducing,  we  get 


the  g«ate.t  range  =:^-j^j. 


(*) 


1S7.  The  «agl«  fti  Bltvatloii  uo  th«t  tiui  Partlcio 
nwy  pUM  through  a  Given  Point— From  Art.  153, 
there  are  two  directions  in  which  a  particle  may  be  pro^ 
jected  so  as  to  reach  a  given  point;  and  they  are  equally 

inclined  to  the  direction  of  projection  («  =  -). 

Let  the  given  point  lie  in  the  plane  which  makes  an 
angle  ft  with  the  horizon,  and  Btijipoge  its  abscissa  to  be  A ; 
then  we  must  have  from  (1)  of  Art.  155 

2w» 
^-^^cos«gin(a-/J)=A. 

-f  «'  and  «"  be  the  two  values  of  a  which  satisfy  this 
equation,  we  must  have 

ooi  «'  sin  (•'  -  /I)  =  008  «"  sin  («"  ~  0) ; 


■"■•P|*|. 


IIL 


iT 


27a      EQUATION  OF  TBAJBCTOST,  SECOND  METHOD. 


and  therefore  a"  —  ^  =  5  —  «', 


I 


i 


or 


»"-ig  +  P)=i(i  +  /')-«' 


(1) 


But  each  member  of  (1)  is  the  angle  between  one  of  the 
directions  of  projection  and  the  direction  for  the  greatest 
range  [Art.  156,  (2)].  Hence,  as  in  Art  152,  the  two 
directions  of  projection  which  enable  the  particle  to  pass 
through  a  point  in  a  given  plane  through  the  point  of  pro- 
jection, are  equally  inclined  to  the  direction  of  projection 
for  the  greatest  range  along  that  plane.  (See  Tait  and 
Steele's  Dynamics  of  a  Particle,  p.  89.) 

158.  Second  Method  of  Finding  the  Equation  of 
the  Trujectory.— By  a  somewhat  simpler  method  than 
that  of  Art.  151,  we  may  find  the  equation  of  the  path  of 
the  projectile  as  the  resultant  of  a  uniform  velocity  and  an 
acceleration  (Art  148). 

Take  the  direction  of  projection  (Fig.  78)  as  the  axis  of 
X,  and  the  vertical  downwards  from  the  point  of  projection 
as  the  axis  of  y.  Then  (Art.  149,  Rem.)  the  velocity,  v, 
due  to  the  projection,  will  carry  the  particle,  with  uniform 
motion,  parallel  to  the  axis  of  x,  while  at  the  same  time,  it 
is  carried  with  constant  acceleration,  g,  parallel  to  the  axis 
of  y.  Hence  at  any  time,  t,  the  equations  of  motion  along 
the  axes  of  x  and  y  respectively  are 

X  —  vt, 

That  is,  if  the  particle  were  moving  with  the  velocity  v, 
alone,  it  would  in  the  time  t,  arrive  at  Q ,  and  if  Jt  were 
then  to  move  with  the  vertical  acceleration  g  alone  it  would 
in  the  same  time  arrive  at  P;  therefore  if  the  velocity  v 


f' 


W  MBTHOP. 


—  a. 


(1) 


etween  one  of  the 
u  for  the  greatest 
Lrt  152,  the  two 
18  particle  to  pass 
the  point  of  pro- 
ition  of  projection 
I.     (See  Tait  and 


the  Equation  of 

pier  method  than 
tion  of  the  path  of 
m  velocity  and  an 

78)  as  the  axis  of 
point  of  projection 
n.)  the  velocity,  v, 
;icle,  with  uniform 
it  the  same  time,  it 
parallel  to  the  axis 
18  of  motion  along 


rith  the  velocity  v, 

Q ,  and  if  It  were 

on  g  alone  it  would 

re  if  the  velocity  v 


EXAMPLES. 


S73 


and  the  acceleration  g  are  simuUaneotis,  the  particle  will  in 
the  time  t  arrive  at  P  (Art.  149,  Rem). 
Eliminating  t  we  have 

g  " 

which  is  the  equation  of  a  parabola  referred  to  a  diameter 

and  the  tangent  at  its  vertex.    The  distance  of  the  origin 

from  the  directrix,  being  J^th  of  the  coefficient  of  y,  is 

V* 

H-,  asm  Art.  163,  (8). 

BXAMFLES. 

1.  From  the  top  of  a  tower  two  particles  are  projected  at 
angles  a  and  /3  to  the  horizon  with  the  same  velocity,  v,  and 
both  strike  the  horizontal  plane  passing  tlirough  the  bot- 
tom of  the  tower  at  the  same  point;  find  the  height  of 
the  tower. 

Let  A  =  the  height  of  the  tower;  v  =  the  velocity  of 
projection ;  then  if  the  particles  are  projected  from  the 
edge  of  the  top  of  the  tower,  and  x  is  the  distance  from  the 
bottom  of  the  tower  to  the  point  where  they  strike  the 
horizontal  plane  we  have  from  (3)  of  Art.  151 


-  A  =  a;  tan  b  - 1^  (1  +  tan»  a), 


A  =  X  tau  /3  -  g  (1  +  tan»  /3), 


(1) 
(2) 


by  subtraction 

X  = 


2t»» 


Sw*  cos  «  cos  i9  ^ 

g  (tan  a  +  tanT^  ""  ^sin  (a  +  /3)  ' 

which  in  (1)  or  (2)  gives 

k  —  ^^  ^'^^  "  "*^"  ^  ^^  ^^  "*"  ^^ 


■■i 


AlWi.SWl.i'Sirl 


^■?su^i«<  eiis^^T^ss^s^Sie^E^aSj^^Si^^ 


wmm 


H 


i 


rf^ 


874 


vuLooirr  of  mscuARoE  of  skslls. 


2.  Piurticles  are  projected  with  a  giTen  veiocitj  in  all 
lines  in  a  Tertioal  plane  £rom  the  point  0;  it  ia  required  to 
find  the  loona  of  their  highest  points. 

Let  {x,  y)  be  the  highest  point ;  then  from  (2)  and  (3)  of 
Art  162,  we  have 


«*  sin  o  cos  rt 

g-    ; 


»»  Bin»  a 


therefore      sin*  a  =  -^,  and  coa*  «  =  —--• 
1^  2v^y 


g  ' 


ff 


Adding  4^  +  a:* 

which  is  the  equation  of  an  ellipse,  wliose  major  axis  = 

and  t,he  minor  axis  =  „-;  and  the  origin  is  at  the  extremity 
of  the  minor  axis. 

8.  Find  the  angle  of  projection,  a,  so  that  the  area  con- 
tained between  the  path  of  the  projectile  and  the  hori- 
zontal line  may  be  a  muximnm,  and  find  the  value  of  the 
maximum  area. 


Ans.  tt  =  60°  and  Max.  Area 


(3)*. 


4.  Find  the  ratio  of  the  areas  A,  and  A,  of  the  two 

parabolas  described  by  projectiles  whose  horizontal  ranges 

are  the  same,  and  the  angles  of  projection  are  therefore 

complements  of  each  other.  ^       A,        ,     . 

Ana.  -jT^  =  tan'  a. 
At 

159.  Velocity  of  Dischaxge  of  Balls  and  Shells 
from  the  Mouth  of  a  GNul— As  the  result  of  numerous 


SHSLLS. 


n  velocity  in  all 
it  ia  required  to 

■om  (2)  and  (3)  of 


major  axis  =  -  ; 
8  at  the  extremity 


hat  the  area  con- 
ic and  the  hori- 
i  the  value  of  the 


rea 


»ff 


.  (3)*- 


1  Aj  of  the  two 
horizontal  ranges 
ion  are  therefore 

-r^  =  tan'  a. 
A, 

ills  and  Shells 

lult  of  numerous 


m 


ANGULAR   VELOCITY. 


2T5 


experiments  made  at  Wcolwich,  the  following  formula  was 
regarded  as  a  correct  expression  fov  the  velocity  of  balls  and 
shells,  on  quitting  the  gun,  and  fired  with  moderate 
charges  of  powder,  from  the  pieces  of  ordnance  commonly 
used  for  military  purposes : 

t;  =  1600>^^, 

where  v  is  the  velocity  in  feet  per  second,  P  the  weight  of 
the  charge  of  powder,  and  W  the  weight  of  the  ball. 

For  the  investigation  of  the  path  of  a  projectile  in  the 
atmosphere,  see  Chap.  I  of  Kinetics. 

160.  Angular  Velocity,  and  Angnlar  Aocelera< 
tion. — Hitherto  the  method  of  resolving  velocities  and 
wcelerations  along  two  rectangular  axes  has  been  employed. 
It  remains  for  us  to  investigate  the  kinematics  of  a  particle 
describing  a  curvilinear  path,  from  another  point  of  view 
and  in  relation  to  another  system  of  reference.  Before  we 
consider  velocities  and  accelerations  in  reference  to  a 
system  of  polar  co-ordinates,  it  is  necessary  to  enquire  into 
a  mode  of  measuring  the  angular  velocity  of  a  particle. 

Angular  Velocity  way  be  defined  as  the  rate  of  angular 

motion.     Thus  let  (r,  0)  be  the  position  of  the  point  P,  and 

suppose    that  the  radius  vector  has  revolved    uniformly 

through  the  angle  9  in  the   time  t,  then  denoting  the 

angular  velocity  by  w,  we  shall  have,  ae  in  linear  velocity 

(Art.  8) 

6 


a  = 


t 


If  however  the  radius  vector  doe-a  not 
revolve  uniformly  through  the  angle  6 
we  may  always  regard  it  as  revolving 
uniformly  through  the  angle  dd  in  the 
infinitesimal  of  time  dt ;  hence  we  shall 
have  as  the  proper  value  of  w, 


•"aWBWaffihffMl 


276 


SXAMPLSa. 


i    !i 


dB 

dt 


(1) 


Hence,  whether  the  angular  velocity  bo  uniform  or 
variable,  it  is  the  ratio  of  the  anglp  described  by  the  mdius 
vector  in  a  given  time  to  the  time  in  which  it  is  described ; 
thus  the  increase  of  the  angle,  in  angular  velocity,  takes 
the  place  of  the  increase  of  the  distance  frcm  a  fixed  point, 
in  linear  velocity,  (Art  8). 

Angular  Acceleration  is  the  rat-  of  tncretse  of  angular 
velocity  :  it  is  a  velocity  increment,  and  is  measnred  in  the 
same  way  as  linear  accelerafion  (Art.  10).  Thus,  whether 
the  angular  acceleration  is  uniform  or  variable,  it  may 
always  be  regarded  as  uniform  during  the  infinitesimal  of 
time  dt  in  which  time  the  increment  of  the  velocity  will  be 
du.  Hence  denoting  the  angular  acceleration  at  any  time, 
t,  by  (p,  we  have 


.       dot      d  /de\,        .,. 


d£*' 


(2) 


and  thus,  whether  the  increase  of  angular  velocity  is 
uniform  or  variable,  the  angular  acceleration  is  the  increase 
of  angular  velocity  in  a  unit  of  time. 

The  following  examples  are  illustrations  of  the  preceding 
mode  of  estimating  velocities  and  accelerations. 

EXAMPLES. 


1.  If  a  particle  is  placed  on  the  icvolving  line  at  the 
distance  r  from  the  origin,  and  the  line  revolves  with  a 
uniform  angular  velocity,  w,  the  relation  between  the  linear 
velocity  of  the  particle  a&d  the  angular  velocity  may  thus 
be  found. 


•^mmm"- 


BgWWiWgllJliJ 


EXAMPLES. 


m 


(1) 

bo  uniform  or 
ed  by  the  rudius 
h  it  is  described ; 
IT  velocity,  takes 
;m  a  fixed  point, 

rease  of  angular 
measured  in  the 
Thus,  whether 
variable,  it  may 
I  infinitesimal  of 
!  velocity  will  be 
ion  at  any  time, 


(2) 

:nlar  velocity  is 
•n  is  the  increase 

of  the  preceding 
ons. 


ring  line  at  the 
revolves  with  a 
itween  the  linear 
locity  may  thus 


( 


Let  dd  be  the  angle  through  which  the  radius  revolves  in 
the  time  dt,  and  let  ds  be  the  path  described  by  the  particle, 
so  that  ds  =  rdO ; 


then 


<l8 


de 


dt  =  ''di  =  "'"' 


so  that  the  linear  velocity  varies  as  the  angular  velocity  and 
the  length  of  the  radius  jointly. 

2.  If  the  angular  acceleration  is  a  constant,  as  ^ ;  then 
from  (2)  we  have 

d*e     ^ 
di»='^' 

•••    Ji  = 'pi  ^  <->of 


and 


e  =  i<t>p  +  ..,#  +  e„ 


where  «„  and  6^  are  the  initial  values  of  w  and  9. 

Hence  if  a  line  revolves  from  rest  with  a  constant  angular 
acceleration,  >.  e  have 

and  the  angle  described  by  it  varies  as  the  square  of  the 
time. 

3.  If  a  particle  revolves  in  a  circle  uniformly,  and  its 
place  is  continually  projected  on  a  given  diameter,  the 
linear  acceleration  along  that  diameter  varies  directly  as 
the  distance  of  the  projected  place  from  the  centre. 

Let  6)  be   the  constant  angular  velocity,  6  the  angle 
between  the  fixed  diameter  and  the  radius  drawn  from  the 
centre  to  its  place  at  the  time  t,  x  the  distance  of  this 
projected  place  from  the  centre.      Then,   calling  a  the  i 
radius  of  the  circle,  we  have 

it  =  «  cos  d. 


St^w"- 


s  t. 


S78 


HnKEnRRSMMSNfT^iB!! 


SXAMPLB8. 

TT  =  —asm  6^  =  — ousinO; 


dt 


dh:  M 

which  proved  the  theorem. 

4.  If  the  angular  acceleration  varies  afi  the  angb 
generated  from  a  given  fixed  lipe,  and  is  negative,  find  the 
angle. 

Here  the  equation  which  expresses  the  motion  is  of  the 
form 


1*0. 


Calling  a  the  initial  value  of  0  we  find  for  the  result 
0  =  a  COB  kt. 

6.  If  a  particle  revolves  in  a  circle  with  a  uniform 
velocity,  ehow  tliat  its  angular  velocity  about  any  point  in 
the  circumference  is  also  uniform,  and  equal  to  one-half  of 
what  it  is  about  the  centre. 

At  present  this  is  sufficient  for  the  general  explanation 
of  angular  velocity  and  angular  accoieration.  We  shall 
return  to  the  subject  in  Chap.  7,  Fart  IIL,  when  we  treat 
of  the  motion  of  i-igid  bodies. 

161.  The  Component  Aooelerationa,  at  any  Jlnstant, 
Jbiaajg,  and  Perpendioolar  to  the  RadiiM  Voctor.— 

Let  {r,  e)  (Fig.  79)  be  the  place  of  the  moving  particle,  i\ 
at  the  time  t,  (x,  y)  being  its  place  referred  to  a  system  of 
rectangular  axes  having  the  same  origin,  and  the  a;«xis 
coiucident  with  the  initial  line.    Then 


a;  s  r  CO8  0;  y  =  r  sin  0; 


11 


WM 


mmm 


sin  0; 


60   M   the   aiigb 
negative,  find  the 

motion  is  of  the 


for  the  reault 


with  a  uniform 
tbout  any  point  in 
ual  to  one-half  of 

metal  explanation 
ation.  We  shall 
[L,  when  we  treat 


I,  at  any  iinstant, 
adiua  Vector.— 

oving  particle,  P, 
red  to  a  syetem  of 
Q,  and  the  x-axia 


(1) 


SAUICAL  AND  TRANSVERSAL  ACCBL8RATI0N8.    279 

m 


therefore  tj-  rs  ^j  co8  ff  —  ."  ?tn  tf  tt  ; 

and 


_  rd^r  _      

'd^~\M       ^\di 


d^i 


-»-[»l5  +  'S]-»<') 


cPx      rtfir 
Similarly 

which  are  the  accelerations  i  arallel  to  the  axes  of  x  and  y. 
Resolving  these  along  the  r«diu8  vector  by  multiplying  (3) 
and  (4)  by  cos  6  and  sin  8  respectively,  since  accelen-ationa 
may  lie  resolved  and  compounded  along  any  line  the  same 
as  velocities  (Art.  149),  and  adding,  we  have 


^cosfl^r  J8ma  =  -^-^ 


m^ 


(«) 


which  is  the  acceleration  along  the  radius  vector.  * 

Multijil>iug  (3)  and  (4)  by  sir.  6  and  cos  6  respectively, 
and  subtracting  the  former  from  the  latter,  we  get 


OOS0 


d*x   ..       ^dr    dd         m 
-dfi''''^  =  ^di'dt-^*^Jfi 


_1  rf/.d<?\. 

-idiVJtr 


(6) 


which  is  the  acceleration  perpendicular  to  the  radius  vector.\ 

163.  The  Component  Aceelerationa,  at  any  in- 
stant, Along,  and  Perpendicular  to  the  TanA^nu   - 

liCt  (jj,  y)  (Pig.  79)  he  the  place  of  the  moving  particle,  P, 
at  the  time  t,  and  «  the  length  of  the  arc  described  during 

•  Bomettntw  called  tht  roMal  ac»eltrtUiim. 
t  BomatfaBM  oalkid  fft*  trmmemal  4UMkfalHm. 


mi' 


'•■i 

hi- 


^1 


' ,  «■ 


»0 


rAifOJSj(rr/aji  acceleratiqii. 


that  time.   Then  the  accelerations  along  the  axes  of  a?  and  y 

(fix        ,  <Pv  da:  du 

•"■^  wfrt  **""  j^ ;  and  the  direction  cosines*  are  ~r-  and  -?• 
"*  B"  as         as 

To  find  the  acceleration  along  the  tangent  we  must  multi- 
ply these  axial  accelerations  by  t-  and  ™,  respectively,  and 
t^d.     Thus  the  tangential  acceleration,  T,  is 


cPx    dx 
diP'ds 


T  =  ^.~  + 


d»y    dy 
d'fl  '  ds' 


(1) 


Sinco  rfs»  =  (ie»  -{  rfys,  therefore,  by  differentiation  we 
have 

ds  d^s  =  dx  dh)  -^  djf  dhf  ; 

and  dividing  by  ds  dP  we  get 
d's       d*x    dx 


dfl 
which  in  (1)  gives 


d»« 


T  = 


dt*' 


(2) 


for  the  acceleration  along  the  tangent. 

Similarly  we  have  for  the  normal  acceleration,  N, 

_^    ^_dh:    dy 
dfl  '  ds  ~  di'i  '  ds 

1    <i8* 
~  P  ■  d/» '  ^'^^  ^**  *'  P*  ^**'  ^*lcnlu8), 

where  p  is  the  radius  of  curvature  ; 


*  CosineaofUwMiglw  wUoliUuiUnKWitniakMwItbUMUMofaudy. 


•|  II  •  im  miwi 


TOS. 

he  axes  of  x  and  y 

^        dx      _,  dti 
!«*  are  ~j-  and  -#• 
as  ds 

t  we  must  multi- 
,  respectively,  and 
',  is 

(1) 

differentiation  we 


(2) 


ration,  N, 


it* 

.  144,  dalcalns). 


Um  um  of  at  wad  y. 


mmmmma 


NORMAL  ACOBLBRATtON. 


V* 

"  =  ?• 


881 
(8) 


if  V  is  the  velocity  of  the  particle  at  the  point  («,  y). 

Hence  at  any  point,  P,  of  the  trajectory,  if  the  accelera- 
tion is  resolved  along  the  tangent  to  the  curve  at  P  and 
along  the  normal,  the  accelerations  along  the  two  lines  arc 
respectively 

d^ 


and    -• 
9 


163.  When  the  Acceleration  Z''erpendionlar  to 
the  Rad5,ii8  Vector  is  sero. — Then  from  (6)  of  Art.  161 
we  have 

r*  T-  =  constant  =  h  suppose ; 


dd 

h 

•'•    dt  - 

r* 

> 

dr 

dr    dd 

h 

dr 

dt 

~  d)'  dt 

•— 

r> 

de* 

and 


'•*     dt*  ~  r*'  de»      VW' 
which  in  (6)  of  Art.  161  gives 
the  acceleration  along  the  radius  vector 

'-~t*dm      ^r»\dsi}      f«' 
an  expression  which  is  independent  of  t. 


(l) 


This  may  be  put  into  a  more  convenient  form  as  follows: 


I«t  f  =  - ;  then 


dr 

de 


du 

de' 


«mm. 


28!^ 


CONSTANT  ANeULAB   VBLOCITT. 


tPr 


—  1     ^  J.  ^  l^^ 


which  in  {1)  and  reducing,  gives 

the  acceleration  along  the  radias  vector 


(2) 


From  these  two  forranlse  the  'iaw  of  acceleration  along 
the  radias  vector  may  be  deduced  when  the  curve  is  given, 
and  the  curve  may  be  deduced  when  the  law  of  accelera- 
tion along  the  radius  vector  is  given.  Examples  of  these 
processes  will  be  given  in  Chap.  (2),  Part  III. 

164.  When  th«  Angular  Velocity  is  Constant- 
Let  the  angular  velocity  be  constant  =  «  suppose.    Then 


=  w; 


therefore  from  (5)  of  Art  161 

the  acceleration  along  the  radius  vector 

d*r  , 

The  acceleration  perpendicular  to  the  radius  vector 

dr 


=  2u 


dr 


(1) 


(2) 


and  both  of  these  are  independent  of  ft 

The  following  example  is  an  illustration  of  these 
formulte : 

A  particle  describes  a  path  r7ith  a  constant  angular 
velocity,  and  without  acceleration  along  the  radius  vector ; 
find  (1)  the  equation  of  the  path,  and  (2)  the  acceleration 
perpendicular  to  the  radius  vector. 


"^*»-, 


mmt 


acirr. 
a 


(2) 

acceleration  along 
;he  curve  is  given, 
0  law  of  accelera- 
ilxamples  of  these 
III. 

is  Constant. — 

►  suppose.    Then 


adins  vector 


(1) 


(2) 


itration   of  these 


constant  angular 
the  radius  vector ; 
I  the  acceleration 


j(BaWtg!iKWi'tni|»JIWI|i|IWiWILIWIIMIIII»^ 


COMITAA-T  ANOULAR   VELOCITY, 


(1)  From  (1)  we  have,  from   the   conditions   of   the 
question. 


dp 


—  w»r  =  0. 


Integrating  ve  have 


dr' 


if  r  =  a  when  jj  =  0. 
at 


Therefore 


dr 


—  udt; 


if  r  =  o  when  t  =  0, 


r  =  ^ie^  +  <r-0. 


(8) 


d6 


Also,  as  -^f  r=  u,  therefore  d  =  W,  if  fl  =  0  when  /  =  0. 


Substituting  this  value  of  ut,  we  have, 


which  is  the  path  described  by  the  particle. 


(*y 


(i)  Let  Q  be  the  required  acceleration  perpendicular  to 
the  radius  veotor;,  then  from  (2)  wo  have 

I 

==  a«i  («-«  _  «--*),  from  (8) 


Bte— 


984 


KXAMPLES. 


aw*  (e  —  «-•) 
2a*»(r»  — a»)*; 


(6) 


which  is  the  acceleration  perpendicnlar  to  the  radias 
vector. 

The  preceding  discussion  of  Kinematics  is  safBcicnt  for 
this  work.  There  are  various  other  problems  which  might 
bo  studied  as  Kinematic  questions,  and  inserted  here  ;  but 
we  prefer  to  treat  them  from  a  Kinetic  point  of  view. 

For  the  investigation  of  the  kinematics  of  a  particle 
describing  a  curvilinear  path  in  space,  see  Price's  Anal. 
Mech's,  Vol.  I,  p.  430,  also  Tait  and  Steele's  Dynamics  of 
a  Particle,  p.  12. 

EXAMPLES. 

1.  A  particle  describes  the  hyperbola,  a;y  =  if ;  find  (1) 
the  acceleration  parallel  to  the  axis  of  z  if  the  velocity 
parallel  to  the  axis  of  ^  is  a  constivnt,  /3,  and  (2)  find  the 
acceleration  parallel  to  the  axis  of  y  if  the  velocity  parallel 
to  the  axis  of  x  is  a  constant,  a. 

Ans.  (l)^a*;(2)^V. 

2.  A  particle  describes  the  parabola,  y*  =  ^ax ;  find 
the  acceleration  parallel  to  the  axis  of  y  if  the  velocity 
parallel  to  the  axis  of  a;  is  a  constant,  o.       ^  . .         4a*a' 


Ans. 


f 


3.  A  particle  describes  the  logarithmic  curve,  y  =  of; 
find  (1)  the  z-component  of  the  acceleration  if  the  y-com- 
ponent  of  the  velocity  is  a  constant,  /3,  and  (2)  find  the 
y-component  of  the  acceleration  if  the  x-component  of  the 
velocity  is  a  constant,  «. 

<^>-^^ga'<*^)«*^^''8''>'y- 


Ans. 


JBXAMPLBA 


386 


) 

^;  (5) 

*r   to   the  radins 

M  is  Bnfficient  for 
lema  which  might 
[iserted  here;  but 
>int  of  view, 
.ties  of  a  particle 
see  Price's  Anal, 
ele's  Dynamics  of 


y  =  /fc»;  find  (1) 

X  if  the  velocity 

and  (2)  find  the 

}  velocity  parallel 

'*•;  (2)  ^V. 

f  =  iax;   find 
f  if  the  velocity 


Ans. -- 


carve,  y  :=:  of; 
3n  if  the  y-com- 
»nd  (2)  find  the 
imponent  of  the 

!)a»  (log  a)«y. 


4.  A  particle  describes  the  cycloid,  the  starting  point 
being  the  origin;  find  (1)  the  a^component  of  the  accel- 
eration if  the  y-component  of  the  velocity  is  (i,  and  (2)  find 
the  y-component  of  the  acceleration  if  the  a;-componont  of 
the  velocity  is  o.  ^^ .    ^^^       ffl«y       .  ^jj)  _  ^. 


Ata,  (1) 


(2«y-y)*'  ^ 


5.  A  particle  describes  a  catenary,  y  =  s(«*'  +  e   "1; 

find  (1)  the  a;-component  of  the  acceleration  if  the  y-com- 
ponent  of  the  velocity  is  /3,  and  (2)  find  the  y-component 
of  the  acceleration  if  the  aMsomponent  of  the  velocity  is  «. 


Am.  (1) 


6.  Determine  how  long  a  particle  takes  in  moving  from 
the  point  of  projection  to  the  farther  end  of  the  latus 

rectum.  ^       **  /  •         ,  \ 

Ans.  -  (sm  a  +  cos  a). 

9 

7.  A  gun  was  fired  at  an  elevation  of  60°;  the  ball 
struck  the  ground  at  the  distance  of  2449  ft. ;  find  (1)  the 
velocity  with  which  it  left  the  gun  and  (2)  the  time  of 
flight.     (y  =  32i). 

Ana.  (1)  200  ft.  per  sec;  (2)  19.05  sees. 

8.  A  ball  fired  with  velocity  u  at  an  inclination  a  to  the 
horizon,  just  clears  a  vertical  wall  which  subtends  an  angle, 
(i,  at  the  point  of  projection;  determine  the  instant  at 
which  the  ball  just  clears  the  wall. 

«  sin  a  —  \gt  _ 


Ana. 


u  cos  a 


im(3. 


9.  In  the  preceding  example  determine  the  horizontal 
distance  between  the  foot  of  the  wall  and  the  point  where 


the  ball  strikes  the  ground. 


Ana.  —  co8»  «  tan  /3. 
8 


vT 


386 


BXAMPLSa, 


10.  At  the  diatanco  of  a  quarter  of  a  mile  from  the  bot- 
tom of  a  oUff,  which  ia  120  ft.  high,  a  shot  ia  to  be  fired 
which  ahall  just  clear  the  clifiF,  and  paaa  over  it  horizon- 
tally ;  find  the  angle,  a,  and  velocity  of  projection,  v. 

Ant.  a  =  10°  18';  t>  =  490  ft.  per  aec. 

11.  When  the  angle  of  elevation  ia  40°  the  range  is 
2449  ft. ;  find  the  range  when  the  elevation  is  29|°. 

Ana.  2131.6  ft. 


12.  A  body  is  projected  horizontally  with  a  velocity  of 
4  ft.  per  see. ;  find  the  latua  rectum  of  the  parabola  de- 
Bcribed,  (g  =  32).  Ana.  1  foot. 

13.  A  body  projected  from  the  top  of  a  tower  at  an  angle 
of  45°  above  the  horizontal  direction,  fell  in  5  seca.  at  a 
distance  from  the  bottom  of  the  tower  equal  to  its  altitude ; 
find  the  altitude  in  feet,  (g  =  32).  Ana.  200  feet 

14.  A  ball  ia  fired  up  a  hill  whoae  inclination  ia  15°; 
the  inclination  of  the  piece  is  45°,  and  the  velocity  of  pro- 
jection is  500  ft.  per  sec. ;  find  the  time  of  flight  before 
it  strikes  the  hill,  and  the  distance  of  the  place  where  it 
falls  from  the  point  of  projection.* 

Ana.  T  =  16.17  aecs.;  R  =  1.121  miles. 

15.  On  a  deacending  plane  whose  inclination  is  12°,  a 
ball  fired  from  the  top  hits  the  plane  at  a  distance  of  two 
miles  and  a  half,  the  elevation  of  the  piece  ia  42°  ;  find  the 
velocity  of  projection.  Ana.  v  =  579.74  ft.  per  aec. 

16.  A  body  ia  projected  at  an  inclination  a  to  the  hori- 
zon; determine  when  the  motion  is  perpendicular  to  a 
plane  which  is  inclined  at  an  angle  /3  to  the  horizon. 

u  sin  a 


Atu. 


u  COB  a 


^  =  ±oot|9. 


*  The  nnge  on  the  Inclined  plane. 


mile  from  the  bot- 
Bbot  is  to  be  fired 
)8  over  it  horiaon- 
>rojectiun,  v. 
490  ft.  per  sec. 

40"  the  range  is 
m  is  29|°. 
Ans.  2131.5  ft, 

ffith  a  velocity  of 

'.  the  parabola  de- 

Ans.  1  foot. 

tower  at  an  angle 
ill  in  5  sees,  at  a 
lal  to  its  altitude ; 
Ans.  800  feet. 

iclination  is  15°; 
3  velocity  of  pro- 
5  of  flight  before 
e  place  where  it 

=  1.121  miles. 

ination  is  12°,  a 
.  distance  of  two 
is  42°  ;  find  the 
L74  ft.  per  sec. 

1  a  to  the  hori- 
rpendicutar  to  a 
e  horizon. 

^'  =  ±  oot  /3. 


nHMmii 


MXAMPLS& 


287 


17.  Calculate  the  maximum  range,  and  time  of  flight, 
on  a  descending  plane,  the  angle  of  depression  of  which  w 
15°,  the  velocity  of  \   ejection  being  1000  ft,  per  sec. 

Ana.  Max.  range  =  7.98  miles  ;  T  =  51.34  sec. 

18.  With  what  velocity  doe^  the  ball  strike  the  plane  in 
the  last  example  ?  Ans.  V  =  1303  feet. 

19.  If  a  ship  is  moving  hori«on tally  with  a  velocity 
=  3</,  and  a  body  is  let  fall  from  the  top  of  the  mast,  find 
its  velocity,  V,  and  direction,  d,  after  4  sees. 

Ans.  Y  —  5g;  e  =  tan-»  f . 

20.  A  body  is  projected  horizontally  from  the  top  of  a 
tower,  with  the  velocity  gained  in  falling  down  a  space 
eqnal  to  the  height  of  the  tower ;  at  what  distance  from 
the  base  of  the  tower  will  it  strike  the  ground  ? 

Ana.  R  =  twice  the  height  of  the  tower. 

21.  Find  the  velocity  and  cime  of  flight  of  a  body  pro- 
jected from  one  extremity  of  the  base  of  an  equilateral 
triangle,  and  in  the  direction  of  the  side  adjacent  to  that 
extremity,  to  pass  through  the  other  extremity  of  the  base. 


Ans. 


»  =  \/f> 


T  =  '^?'* 


3  '  -  9 

2'4.  Given  the  velocity  of  sound,  V;  find  the  horizontal 
range,  when  a  ball,  at  a  given  angle  of  elevation,  «,  is  so 
projected  towards  a  person  that  the  ball  and  sound  of  the 
discharge  reach  him  at  the  same  instant 

Ans.  —  tan  «. 
ff 

23.  A  body  is  projected  horizontally  with  a  velocity  of 
4g  from  a  point  whose  height  above  the  ground  is  16jr ;  find 
the  direction  of  moHon,  d,  (1)  when  it  has  fallen  half-way 
to  the  ground,  and  (2)  when  half  the  whole  time  of  falling 


has  elapsed.  ^^ 


(l)e  =  46°;  (2)  e  =  tan-» -^-=. 


288 


EXAMPLBS. 


I,-. 

I;  :• 


24.  Particles  are  projected  with  a  given  velocity,  v,  in 
all  lines  in  a  vertical  plane  from  the  point  0  ;  find  the  locus 
of  them  at  a  given  time,  /. 

Ans.  a:*  +  (y  +  io^Y  —  ^^>  which  is  the  equation  of  a 
circle  whose  radius  is  vt  and  whoso  centre  is  on  the  axis  of 
y  at  a  distance  ^P  below  the  origin. 

25.  How  much  powder  will  throw  a  13-inch  shell* 
40G0  ft.  on  an  inclined  plane  whose  angle  of  elevation  is 
10°  40' ;  the  elevation  of  the  mortar  being  35". 

Ans.  Charge  =  4.67  lbs. 

26.  A  projectile  is  discharged  in  a  horizontal  direction, 
with  a  velocity  of  450  ft.  per  sec,  from  the  summit  of  a 
conical  hill,  the  vertical  angle  of  which  is  120° ;  at  what 
distance  down  the  hillside  will  the  projectile  fall,  and  what 
will  be  the  time  of  flight? 

Ans.  Distance  =  2812.5  yards;  Time  =  16.23  sees. 

27.  A  gun  is  placed  at  a  distance  of  500  ft.  from  the  base 
of  a  cliff  which  is  290  ft.  high  ;  on  the  edge  of  the  cliff 
there  is  built  the  wall  of  a  castle  60  ft.  high ;  And  the 
elevation,  «,  of  the  gun,  and  the  velocity  of  discharge,  v, 
in  order  that  the  ball  may  graze  the  top  of  the  castle  wall, 
and  fall  120  ft.  inside  of  it 

Ans.  «  =  63°  19' ;  v  =  165  ft.  per  sec. 

28.  A  piece  of  ordnance  burst  when  50  yards  from  ^-^ 
wall  14  ft.  high,  and  a  fragment  of  it,  originally  in  con- 
tact with  the  groand,  after  grazing  the  wall,  fell  6  ft. 
beyond  it  on  the  opposite  side ;  find  how  high  it  rose  in 
the  air.  Ans.  94  ft. 


*  The  weight  of  •  la-inch  ihell  U  IM  Ibe. 


Ire : 

m 


mi 


iven  velocity,  v,  in 
ifcO;  find  the  locus 

!  the  equation  of  a 
e  is  on  the  tuis  of 


a    13-inch    shell* 

gle  of  elevation  is 

g35". 

irge  =  4.67  lbs. 

rizontal  direction, 
the  summit  of  a 
is  120° ;  at  what 

tile  fall,  and  what 


le  =  16.23  sees. 

[)  ft  from  the  base 
edge  of  the  cliff 
t  high;  find  the 
7  of  discharge,  v, 
)f  the  castle  wall, 

165  ft.  per  sec. 

50  yards  from  ^ 
originally  in  con- 
B  wall,  fell  6  ft. 
»w  high  it  rose  in 
An8.  94  ft. 

Iba. 


PART    III. 
KINETICS  (MOTION  AND  FORCE). 


CHAPTER    I. 

LAWS  OF  MOTION— MOTION  UNDER  THE  ACTION  OF 
A  VARIABLE  FORCE— MOTION  IN  A  RESISTING 
MEDIUM. 

165.  Deflnitioiui. — Kinetics  is  that  branch  of  Dynamics 
which  treats  of  the  motion  of  bodies  under  tfie  action  of 
forces. 

In  Part  I,  forces  were  considered  with  reference  to  the 
pressures  which  they  produced  upon  bodies  at  rest  (Art. 
16),  i.  e.,  bodies  under  the  action  of  two  or  more  forces 
in  equilibrium  (Art  26).  In  Part  II  we  considered  the 
purely  geometric  properties  of  the  motion  of  a  point  or 
particle  without  any  reference  to  the  causes  producing  it, 
or  the  properties  of  the  thing  moved.  We  are  now  to 
consider  motion  with  reference  to  the  causes  which  produce 
it,  and  the  things  in  which  it  is  produced. 

The  student  must  here  review  (Chapter  I,  Part  I,  and  obtain  clear 
conceptions  of  MomeiUutn,  Aeeelsrattini  tf  Mimuintum,  and  the  KinHie 
meamre  of  Fane  (Arts.  18. 14. 19,  and  90),  as  this  is  necessary  to  a  fall 
understanding  of  the  fundamental  laws  of  motion,  on  the  trutli  of 
which  all  our  succeeding  investigations  are  founded. 

166.  Newton's  Laws  of  Motioa— The  fundamental 

13 


1' 


\ 


■1. 


t. 


mm 


290 


A^rrojvs  LAWS  or  motion. 


:•  I 


: 


principles  in  aooordancfi  with  wliich  motion  takes  pliicc  are 
embodied  in  three  statements,  generally  known  as  New/on^ 
Laws  of  Motion.  These  laws  must  be  considered  as  resting 
on  convictions  drawn  from  observation  and  experiment, 
and  not  on  intuitive  perception.*  The  laws  are  the  fol- 
lowing: 

Law  \.— Every  body  continues  in  its  state  of  rest 
or  of  uniform  motion  in  a  straight  line,  except  in 
so  far  as  it  is  compelled  hy  force  to  change  that 
state. 

Law  II.— Change  of  motion  is  proportional  to  the 
force  applied,  and  takes  place  in  the  direction  of 
the  straight  line  in  which  the  force  acts. 

Law  IIL— 2b  every  action  there  is  always  an 
equal  and  contrary  reaction;  or,  the  mutual  ac- 
tions of  any  two  bodies  are  always  equal  and  oppo- 
sitely directed. 

167.  Remarks  on  Law  l— Uw  i  snppUeB  m  with  » 

definition  of  force.  It  indicates  that  force  is  that  which  tends  to 
change  a  body's  state  of  rest  or  of  uniform  motion  in  a  straight  line  ; 
for  if  a  body  does  not  continue  in  its  state  of  rest  or  of  uniform  mo- 
tion in  a  straight  line  it  mupt  bo  under  the  action  of  force. 

A  body  has  no  power  to  change  its  own  state  as  to  rest  or  motion  ; 
when  it  is  at  rest,  it  has  no  power  of  putting  itself  in  motion  ;  when 
in  motion  it  hr«  no  power  of  increasing  or  diminishing  its  velocity. 
Matter  is  inert  (Art.  8).  If  it  is  at  rert,  it  will  mnain  at  rest ;  if  it  is 
moving  with  a  given  velocity  along  a  rectilinear  path,  it  will  continue 
to  move  with  that  velocity  along  that  path.  It  Is  alike  natural  to 
mat(*r  to  be  at  rest  or  in  motion.  Whenever,  therefore^  a  body's 
state  is  changed  either  from  rest  to  motion,  or  fr<mi  motion  to  rest, 
or  when  its  velocity  is  Increased  or  diminished,  that  change  is  due  to 
some  external  cause.  This  cause  is  called  force  (Art.  16) ;  and  the 
word /WW  is  used  in  Kinetics  in  this  meaning  only. 


*  Thonuoa  and  Tait'i  Nat.  FhU.,  p.  Stt. 


riON. 

ion  takes  place  are 
known  as  Newfon'f 
nsidcred  as  resting 
I  and  exporiment, 
laws  are  the  fol- 


its  state  of  rest 

'  line,  except  in 

to  change  that 

portional  to  the 
•he  direction  of 
;  acta. 

is  always  an 
he  mutual  ac- 
'■qual  and  oppo- 


snpplies  as  with  k 
bat  which  tends  to 
I  in  a  straight  line  ; 
It  ot  of  uniform  rao- 
I  of  force. 

a  to  rest  or  motion  ; 

)lf  in  motion ;  when 

tishing  its  velocity. 

oain  at  rest ;  if  it  is 

tath,  it  will  continue 

is  alike  natural  to 

therefore,  a  body's 

imi  motion  to  rest, 

at  cban(re  is  due  to 

(Art.  16) ;  and  the 

J. 

a. 


itnMAiiKa  oy  law  it. 


291 


168.  Remark*  on  Law  IL — Law  ll  asserts  that  it  any 
force  generates  motion,  a  double  force  will  generate  douljle  motion, 
nnd  so  on,  whether  applied  simultaneously  or  successively,  instan 
tiineously  or  gradually.  And  this  motion,  if  the  body  was  movin|< 
iK'i'orehand,  is  either  added  to  the  previous  motion  if  directly  consj  ir 
ing  with  it,  or  is  -^ubtnwrted  if  directly  opposed  ;  or  is  geometrit-ally 
compounded  with  it  according  to  the  principles  already  explained. 
(Art.  29),  if  the  line  of  previous  motion  and  the  direction  of  tho  forco 
are  inclined  to  each  other  at  an  angle.  The  term  motio;,  here  means 
quantity  of  motion,  and  the  phrase  dutnge  of  motion  here  means  mtc 
of  change  of  quantUy  of  motion  (Art.  14).  If  tho  force  be  finite  It  will 
require  a  finite  time  to  produce  a  iiensible  change  ot  motion,  and  the 
cliange  of  momentum  produced  by  it  will  depend  upon  the  time  dur- 
ing which  it  acts.  The  change  of  motion  must  then  be  understood  to 
be  the  change  of  momentum  produced  per  unit  of  time,  or  the  rate 
of  change  of  momentum,  or  acceleration  of  momentum,  which  agrees 
wHh  the  principles  already  explained  (Arts.  14  and  20).  In  this  law 
uotlilng  is  said  about  tho  actual  motion  of  the  body  before  it  was 
acted  on  by  tho  force ;  it  is  only  the  change  of  motion  that  concerns 
us.  The  same  force  will  produce  precisely  the  same  change  of  mo- 
tion in  a  body ;  whether  the  body  be  at  rest,  or  in  motion  with  any 
velocity  whatever. 

Since,  when  several  forces  act  at  once  or.  ^.  particle  either 
at  rest  or  in  motion,  the  second  law  of  motion  is  true  for 
every  one  of  these  forces,  it  follows  that  each  must  have  the 
same  effect,  in  so  far  as  the  change  of  motion  produced  by 
it  is  concerned,  as  if  ti  were  the  only  force  in  action. 
Hence  the  assertion  of  the  second  law  may  be  put  in  the 
following  form : 

When  any  number  of  forces  act  simultaneously  on  a  body, 
whether  at  rest  or  in  motion  in  any  direction,  each  force  pro- 
duces in  the  body  the  same  change  of  motion  as  if  it  alone 
had  acted  oh  the  body  at  rest. 

It  follows  from  this  view  of  the  law  that  all  problems 
vhich  involve  forces  acting  simultaneously  may  bo  treated 
as  if  the  forces  acted  successively. 

The  operations  of  thia  law  have  already  been  considered  in  Kine- 


IM 


292 


HEM  ARKS  ON  LAW  U. 


I    :    • 


iv 


TDatics  (Art.  14ft) ;  hut  motion  there  was  underotood  to  mean  wHocUy 
only,  since  the  mass  of  the  body  was  not  considered.  Thia  law  in- 
cludes, therefore,  the  law  of  the  composition  of  velocities  already 
referred  to  (Art.  20 j.  Another  consequence  of  the  law  is  the  follow- 
ing :  Since  forces  are  measured  by  the  changes  of  motion  they  i  ro- 
duce,  and  thirT  directions  assigned  by  the  directions  in  which  these 
chnnges  are  produced,  ani  since  the  changes  of  motion  of  one  and  the 
same  body  are  in  the  directions  of,  and  proportional  to,  the  changes 
of  velocity,  therefore  a  single  force,  measured  by  the  resultant  change 
cf  velocity,  and  in  its  direction,  will  bo  the  equivalent  of  any  uuml)er 
of  simultaneously  acting  forcus. 

Hence, 

Hie  remltant  of  any  number  of  concurring  forces  is  to  be 
found  by  the  same  geometric  protxds  as  the  reauUant  of  any 
number  of  simultaneous  velocities,  and  conversely. 

From  this  follows  at  once  the  Polygon  of  Velocities  anrl 
thp  Parallelopiped  of  Velocities  from  the  Poly»joii  and 
Parallelopipcd  of  Forees,  as  was  described  in  Art.  142. 

This  law  also  ^ves  us  the  meniw  of  measuring  force,  and  also  of 
measuring  tho  mats  of  a  body :  for  the  actions  of  different  forces  upon 
the  bame  body  for  oijual  times,  evidently  produce  changes  of  velocity 
which  ore  proporliuitai  to  the  forces.  Also,  if  equal  forces  act  on  dif 
f.jrent  Ixxlios  for  e<iual  times,  the  rhanjres  of  velocity  ])rofluced  must 
Iw  invfi'tfly  as  the  mat»exo(  the  bodies.  Again,  if  different  Ijodies, 
each  acted  on  by  a  force,  acquire  In  the  same  time  the  same  changes 
of  velocity,  the  forces  must  be  proportional  to  the  musses  of  the 
bodies.  This  means  of  measuring  force  is  practically  the  same  us 
tliot  already  deduced  by  abstract  reasoning  (Arts.  19  and  20). 

It  appears  from  thie  law,  tliat  every  theorem  of  Kine- 
matics ctmnceted  with  uccelerat'on  has  its  counterpart  in 
Kinetics.  Thi's,  the  meauure  of  acceleration  or  velocity 
increment,  (Art.  9),  which  wua  discussed  in  Chap.  I  (Arts. 
9  and  10),  and  in  Kinematics  (A:     135),  and   which  is 

(Ps 
denoted  by  /  or  its  equal    ^-j,  is  also  the  effect  and  the 

measure  of  force  ;  therefore  all  the  results  of  the  equation 


itood  to  mean  ndocUy 
lidered,  Thia  law  in- 
of  velocities  already 
the  law  is  the  follow - 
s  of  motion  tliey  jro- 
Btions  in  which  tlicso 
motion  of  one  and  tlif; 
ional  to,  the  changes 
'  the  resultant  change 
vaieat  of  any  number 


ring  forces  is  to  bo 
he  resultant  of  any 
nversely. 

n  of  Velocities  anfl 
the  Poly»joii  and 
I  in  Art.  142. 

ng  fiirce,  and  also  of 
'  difibrent  forces  upon 
3  clmngeR  of  velocity 
iual  forces  act  on  dif- 
iocity  ])ro(luced  mast 
in,  if  different  IwxlirB, 
me  the  same  fhangos 
n  the  musses  of  the 
Helically  the  same  us 
.  19  and  20). 

tlieorom  of  Kine- 
it«  counter})art  in 
ration  or  velocity 
in  Chap.  I  (Arts. 
6),  and   which  is 

lie  effect  and  the 

J  of  the  equation 


H 

RBMARKS  ON  LAW  IT. 

293 

d*8 

(1) 

'■f 

1 

its  various  forms,  and  the  remarks  which  have  been  made 
on  it,  are  applicable  to  it  when  /  is  the  accelerating  force. 
Thus,  (Art.  162),  we  see  that  the  force,  under  which  a 
particle  describes  any  curve,  may  bo  resolved  into  two 
components,  one  in  the  tangent  to  the  curve,  the  other 
towards  the  centre  of  curvature ;  their  magnitudes  being 
tlie  acceleration  of  momentum,  tiud  the  product  of  the 
momentum  into  the  angular  velocity  about  the  centre  of 
curvature,  respectively.  In  the  case  of  uniform  motion, 
the  first  of  these  vanishes,  or  the  whole  force  is  perpen- 
dicular to  the  direction  of  motion.  When  there  is  no  force 
perpendicular  to  the  direction  of  motion,  there  is  no  curva- 
ture, or  the  path  is  a  straight  line. 

Henco  if  we  suppose  the  particle  of  mass  wi  to  be  at  the 
iwint  {x,  y,  z),  and  resolve  the  forces  acting  on  it  into  the 
three  rectangular  components,  X,  Y,  Z,  wo  have 


m 


dt* 


m 


^l  =  Y:m^  =  Z. 


dh 


dC^ 


dl^ 


(2) 


In  several  of  the  chapters  those  equations  will  be  sim- 
plified by  assuming  unity  as  the  maas  of  the  moving 
particle.  When  this  Cannot  be  done,  it  is  sometimes  con- 
venient to  assume  X,  Y,  Z,  as  the  component  forces  on  the 
unit  mass,  and  (8)  becomea 


m 


d*.t 
dfi 


mX,  etc. 


It  will  be  ob- 


from  which  m  may  of  course  be  omitted, 
served  that  an  equation  such  as 

^,  =  X 

may  be  interpreted  either  as  Kino.ical  or  Kinematical ;  if 


.;•!«  ■ 


...ritfM 


IMM 


2H 


aSMASKS  ON  LAW  W. 


%/ 


the  former,  the  uuit  of  mass  must  be  understood  as  a  fac- 
tor on  the  left-hand  ide,  in  which  case  X  is  the  a;-eom- 
ponent,  for  the  unit  oi  mass,  of  the  whole  force  exerted  on 
the  moving  body. 

The  fl.«t  two  lawB,  h»ve,  therefore,  furniahed  us  with  r  definition 
and  a  meaturtof  forte;  and  they  aleo  tsho*  i  ow  to  oompoaud,  aod 
therefore  bow  to  rwjolve,  forcee;  and  also  how  to  investigate  the 
conditions  of  equilibrium  or  motion  of  a  single  particle  subjected  to 
given  forces. 

169.   RemarkB  on  Law  IH— According  to  Law  III.  if  one 

body  presses  or  draws  another,  it  is  pressed  or  drawn  by  this  other 
with  an  wjiial  force  In  the  opposite  direction  (Art.  10).  A  horse 
Vjwing  a  boat  on  a  canal,  is  pulled  backwards  by  a  force  equal  to  that 
which  he  impresses  ou  tho  towiogro[je  fcrwarda.  If  one  body  strikes 
another  body  and  changes  tho  rootiou  of  the  other  body,  its  own 
motion  will  bo  changed  in  an  equal  quantity  and  in  the  opposite 
direclioa ;  for  at  each  instant  during  the  impact  the  bodies  exort  un 
each  other  cqur  1  and  opposite  pressures,  and  ihe  momentum  that  one 
body  loses  is  jMjual  to  that  which  the  other  gains. 

Tlie  earth  atiracts  a  ikUing  pebble  with  a  certain  foiw,  while  the 
pebble  attracu  the  earth  with  an  equal  force.  The  result  is  thot 
while  the  pebble  moves  towards  tho  earth  on  «•<  ..iint  of  ito  attrac- 
tion, th«?  earth  aleo  moves  towards  the  pebble  under  the  influence  o' 
tho  nttruction  of  the  latter ;  but  tlie  mass  of  the  earth  being  enor- 
mously greater  than  that  of  the  pebble  whUe  the  forces  on  the  two 
arising  from  their  mutual  attractions  are  equal,  tho  motion  produced 
thereby  in  the  earth  is  almost  incomparably  leas  than  that  pnxluced 
in  the  pebble,  and  is  consoquintly  insensible. 

It  follows  that  the  sum  of  the  qusni.Mes  of  motion  parallel  to  any 
fixed  direction  of  the  particles  of  any  syrtem  influencing  one  another 
in  any  po.«ible  way,  remains  unchanged  by  their  mutual  action. 
Therefore  if  the  centre  of  gravity  of  any  system  of  mutually 
In'luenciug  iiarticles  is  in  motion,  it  continues  movii.g  unlfomi'v  In  a 
«truiglit  lino,  unless  In  s  >  far  as  the  direction  or  velocity  of  i»  motion 
is  chonged  by  forces  bctwet-n  the  psivicles  and  some  other  matter  not 
Moiiging  to  the  gyHem  :  also  the  centre  of  gn-vity  of  any  system  of 
particles  moves  Just  as  all  the  matter  of  the  system,  if  .».»centrated  in 
a  point,  would  move  under  tho  lntiuenc(>  of  forces  (>*]Ual  ami  parallel 
to  ihe  foreve  ivsHy  miiag  on    its    difll>rent    parts.      (For    furthitr 


-(._^, 


wm 


TWO  LA  Wa  OP  MOTTOIf. 


295 


tideratood  as  a  fac- 
e  X  is  the  a'-com- 
i  force  exerted  on 


as  with  A  definition 
>w  to  oompouud,  aod 
r  to  investigate  the 
particle  subjected  to 


n^  to  Law  III,  if  one 
drawn  hy  tliis  other 
(Art.  10).  A  horse 
a  force  equal  to  that 
If  one  body  strikes 
uther  body,  its  own 
and  in  the  opposite 
i  the  bodle8  exert  un 
momenttim  that  one 

tain  foix^e,  while  the 
The  result  is  that 
W(  oiint  of  its  attrac- 
der  the  influoncii  o* 
e  earth  being  enor- 
le  forces  on  the  two 
he  motion  produced 
than  that  produced 

tinn  parallel  to  any 
ucncing  one  another 
leir  mutual  action. 
yRt«>m  of  mntually 
Hiiir  untform'v  in  a 
alooity  of  i's  motion 
Tie  other  matter  not 
y  of  any  systom  of 
a,  it  ..tincontrated  in 
(Mjual  ami  parallel 
rts.     (For    fortliMr 


remarkr  on  tbese  laws  see  Tait  and  Steele's  Dynunics  of  a  Particle, 
Thomson  and  Tait's  Nat.  Phil.,  Pratt's  Mechanics,  etc.) 

170.  Two  Laws  of  Motion  in  the  Fronch  Troa- 

tises. — Newton's  Laws  of  motion  are  not  adopted  in  the 
principal  French  treatises ;  bet  we  fiud  in  them  two  prin- 
ciples only  as  borrowed  from  experience,  viz.: 

First. — ^The  Law  of  Inertta,  that  a  body,  not  acted 
upon  by  any  force,  would  go  on  for  ever  with  a  uniform 
velocity.    This  coincides  With  Newton's  First  Iaw. 

Second. — That  the  velocity  communicated  is  proportional 
to  the  force.  The  second  and  third  Laws  of  Motion  are 
thus  reduced  to  this  second  priticiple  by  the  French  writers, 
especially  Poisson  aud  Laplace.* 

171.  Motion  of  a  Particle  under  the  Action  of  an 
Attractive  Force. — A  particle  moves  under  a  force  of 
attraction  which  is  in  its  line  of  motion,  and  varies  directly 
as  the  distance  of  the  particle  from  the  centre  of  force;  it  is 
required,  to  determine  the  motion. 

The  point  whence  the  it  fluenco  of  a  force  emanates  is 
called  the  centre  of  force  ;  and  the  force  is  called  an  attrac- 
tive or  a  repulsive  force  according  as  it  attracts  or  repels. 

jjet  0  be  the  centre  of  force,  P  the 
position  of  the  particle  at  any  time,  t,  v  ~ 
its  velocity  at  that  time,  and  let  OP  =  x, 
and  OA  =  a,  where  A  is  the  position  of  the  particle  when 
/  =  0 ;  let  ^  =  the  absolute  force ;  that  is,  the  force  of 
attraction  on  a  unit  of  mass  at  a  unit's  distance  from  0, 
which  is  supposed  to  be  known,  and  is  sometimes  called 
the  strength  of  the  attraction.    At  preaeo*  we  shall  suppose 

*  ParkiDaou'i  MMbaaiei,  p.  187.  See  paper  by  Dr.  Whewell  on  ttip  principle* 
of  DTiuunks.  iMitUcnlarty  as  alalsd  by  FrMieb  wiiten,  Is  tbo  Idinboiili  jonrsal  gf 
Hci*w».  Vol.  VaL 


.-^ 


Fi|.M  M» 


^m 


39i 


aSMAMKS  ON  LAW  Ut. 


the  former,  the  unit  of  mass  must  be  imilerstood  aa  a  fac- 
tor on  the  left-hand  side,  in  which  case  X  ia  the  x-com- 
ponent,  for  the  unit  of  'Tiass,  of  the  whole  force  exerted  on 
the  moving  body. 

The  flnt  two  Uwb,  h»ve,  therefore,  furniahed  aa  with  a  definition 
and  a  meature  of  force ;  aud  they  alau  ahow  how  to  oompouud,  and 
lherefoi«  how  to  reeolve.  forcea;  and  also  how  to  inveetigata  the 
conditiona  of  equilibrium  or  motion  of  a  aingle  particle  aubjectod  to 
given  forces. 


169.  Ztomarka  on  Law  ni.— According  to  Law  III,  if  one 
bodjr  presaea  or  drawa  another,  it  ia  prnsaed  or  drawn  by  thia  other 
with  an  equal  force  in  tlie  oppoeite  directior  (Art.  10).  A  horse 
towing  a  boat  on  a  canal,  is  pulled  backwards  by  a  force  equal  to  that 
whic>i  he  impreasee  ou  the  towing-rope  forwarda.  If  one  l)ody  strikes 
another  Ixhdy  and  changes  the  motion  of  the  other  body,  ita  own 
motion  will  l>o  changed  in  an  equal  quantity  and  in  the  opposite 
dirucUun;  fur  at  each  instant  during  the  impact  the  Ixidiea  exert  <>n 
each  othi>r  oqnal  and  opposite  pressures,  and  the  momentum  that  one 
liody  luBcs  is  equal  to  that  which  the  other  gains. 

The  eorth  attracts  a  falling  pebble  with  a  certain  force,  while  the 
pebble  nttracta  the  eurth  with  an  equal  force.  The  reault  is  that 
while  the  pebble  moves  towards  the  earth  on  account  of  its  attrac- 
tion, the  <>arth  also  moves  towards  the  pebble  under  the  infauenoe  of 
the  nttractitm  of  the  latter ;  but  the  mass  of  the  earth  being  enor- 
WDUiily  greater  than  that  of  the  pebble  while  the  forces  on  the  two 
arising  from  their  mutual  attn  ^tions  are  equal,  the  motion  produced 
thereby  in  the  earth  is  almost  incomparably  leas  than  thai  produced 
iu  the  pebble,  and  is  consequently  insensible. 

It  follows  thai  the  sura  of  the  quantities  of  mai.ion  parallel  to  any 
fixed  direction  of  the  partlclro  of  any  system  influondng  one  another 
in  any  poHsible  Avsy,  remains  unchanged  by  their  mntual  action. 
Therefhro  if  the  c-entro  of  gravity  of  any  system  of  mntnally 
influencing  particles  is  in  motion,  it  oontinuea  movir^|r  uniformly  in  a 
straigRt  lino,  unless  in  bd  fur  us  the  direction  or  velocity  of  i's  motion 
is  changed  by  forcea  between  the  particles  and  some  other  matter  not 
hflonging  to  the  tyntem  :  also  the  centre  of  gravity  of  any  system  of 
]iartirles  moves  Just  as  all  the  matter  of  the  system,  if  concentrated  in 
11  |M>lnt,  would  move  under  the  influenc«<  of  .orres  •■<)ual  and  parallel 
to  ihe  forcva  leaHy  acttHg  on    its    ditlcrr>at    parts.      (For    furtliMT 


'^''flBSfc^i^'fltPilESBfflll 


TWO  LA  wa  OP  Morrox; 


395 


derstood  as  a  fac- 

X  ia  the  ar-com- 

force  exerted  on 


38  with  a  tkfinition 

V  to  eompouud,  »ad 

to  investigats  the 

article  subjected  to 


Of  to  Law  III,  if  one 
Irawn  by  tliia  other 
Art.  16).  A  horse 
I  force  equal  to  that 
If  one  body  atrikea 
ther  body,  its  own 
id  in  the  opposite 
the  IxHiiea  exert  on 
lomentum  that  one 

lin  force,  while  the 
The  result  is  that 
count  of  its  attrac- 
er  the  infauencti  of 
earth  being  enor- 
I  forces  on  the  two 
e  motion  produced 
than  thai,  produced 

nn  parallel  to  any 
pncing  on<»  another 
>ir  mutual  action. 
Rtem  of  routnally 
rfr  uniformly  in  a 
ority  of  i's  motion 
o  other  matter  not 
of  any  systom  of 
.  if  concentrated  in 
Hfual  and  parallel 
ts.     (Por    furtliMf 


remarks  on  these  laws  see  Tait  and  Steele's  Dynamics  of  a  Particle, 
Tikomaon  and  Tait's  Nat  Phil.,  Pratt's  Mechanics,  etc.) 

170.  Two  Laws  of  Motion  in  the  French  Troa- 

tiaoa — Newton's  Laws  of  motion  are  not  adopted  in  the 
principal  French  treatises ;  bnt  we  find  in  them  two  prin- 
ciples only  as  borrowed  from  experience,  viz.: 

First. — ^The  Law  of  Inertia,  that  a  body,  not  acted 
upon  by  any  force,  would  go  on  for  ever  with  a  uniform 
velocity.    ThiS  coincides  with  Newton's  First  T^iw. 

Second. — That  the  velocity  communicated  ia  proportional 
to  the  force.  The  second  and  third  Laws  of  Motion  ai'e, 
thup  reduced  to  this  second  principle  by  the  French  writers, 
especially  Poisson  and  Laplace.* 

171.  Motion  of  a  Pazticto  nnder  tho  Action  of  an 
Attractive  Force. — A  particle  moves  under  a  force  of 
attraction  whtck  is  in  its  line  of  motion,  and  varies  directltf 
as  the  distance  of  the  particle  from  the  centre  nf  force;  it  is 
required,  to  determine  the  motum. 

Tho  point  whence  the  influence  of  a  force  emanates  is 
called  the  centre  of  force  ;  and  the  force  is  called  an  attrac- 
tive or  a  repulsive  force  according  as  it  at/racla  or  repels. 

Ijet  0  be  the  centre  of  forco,  P  the  ^, 
position  of  the  particle  at  any  time,  /.  v 
its  velocity  at  that  time,  and  let  OP  =  x, 
and  OA  =  a,  where  A  is  tho  position  of  the  particle  when 
t  —  0;  let  /i  =  the  absolute  force ;  that  is,  the  force  of 
attraction  on  a  unit  of  mass  at  a  unit's  distance  from  O, 
which  is  supposed  t,o  be  known,  and  is  sometimes  called 
the  strength  of  the  attracti  .i.    At  present  we  shall  suppose 


4-^ 


Ft|.N  M» 


Ail* 


1   n 
9.1 ! 


*  ParWnion'*  Xachanles,  p.  Wl.   See  paper  bj  Dr.  Wkcwell  on  ibe  principle* 
of  Djmamiea,  parttcalarly  •■  ■tated  by  Viench  wrlten,  In  the  Bdlnborgh  joomai  of 
•,  Vol.  VUL 


S96 


A    VARIABLE  ATTRACnVB  tORCU. 


the  maas  of  the  particle  to  be  unity,  aa  it  aimplifles  the 
eqaatiouB.  Then  fix  is  the  magnitude  of  the  force  at  the 
distance  x  on  the  particle  of  unit  mass,  or  it  is  the  accelerr.- 
tion  at  P  ;  and  the  equation  of  motion  is 


(1) 


the  negative  sign  being  taken  because  the  tendency  of  the 
force  18  to  diminish  x\ 


idzcPx 


=  —  %\ix  dx. 


Integrating,  we  get 


d^ 
dp 


=  fi  (a»  -  a^), 


(2) 


if  the  particle  be  at  rest  when  x  =  a  and  t  —  0, 

.'.    -^^===V^dt, 

Va*  -  a? 

the  negative  sign  being  taken,  because  x  decreases  as  t 
increases.  Integrating  again  between  the  limits  correspond- 
ing to  ^  =  /  and  t  =  0, 


COS" 


i?  =  H*/. 


1       -1* 

<  =  -j-  COS  '  -• 


(3) 


Prom  (2)  it  appears  that  the  velocity  of  the  particle  is 
zero  when  a;  =  a  and  -  o  ;  and  is  a  maximum,  viz.:  «^*, 
when  a;  =  0.  Hence  the  particle  moves  ftom  rest  at  A :  its 
velocity  increases  until  it  reaches  0  where  it  becomes  a 


r  rOHCll. 

88  it  simplifies  the 

of  the  force  at  the 

or  it  is  the  accelei^- 

is 


the  tendency  of  the 


(2) 


i<  =  0, 


e  X  decreases  as  t 
e  limits  correspond- 


(3) 


ty  of  the  particle  is 
laximaro,  viz.:  mju*, 
from  rest  at  A ;  its 
here  it  becomes  a 


-. 


-«• 


'jMiaiim'""*  •^I'liiin  'MmxxmsiattKnx^itri/msmmsimieKiiiienis^j 


A    VAMlABm  ATTRACTIVM  Wn^BGJl, 


S97 


masimnm,  and  where  the  force  is  zero ;  the  putiole  passes 
through  that  point,  and  its  velocity  decreases,  and  at  A',  at 
a  distance  =  —  a,  becomes  zero.  From  this  point  it  will 
return,  under  the  action  of  the  force,  to  its  original  posi- 
tion, and  continually  oscillate  over  the  space  2a,  of  which 
O  is  the  middle  point. 
From  (3)  we  find  when  a;  ==  a,  <  =  0  and  when  x  =  0, 

t  =  — T ;  80  that  the  time  of  passing  irom  A  to  0  =  — r . 

and  the  time  from  0  to  A'  is  the  same,  so  that  the  time  of 

oscillation  from  A  to  A'  is  -r*     Tbis  result  is  remarkable, 

as  it  shows  that  the  time  of  oscillation  is  independent  of 
the  velocity  and  distance  of  projection,  and  depends  solely 
on  the  strength  of  the  attraction,  and  is  greater  as  that  is 
less. 

This  problem  includes  the  motion  of  a  particle  within  a 
homogeneous  sphere  of  ordinary  matter  in  a  straight  shaft 
through  the  centre.  For  the  attraction  of  such  a  sphere  on 
a  particle  within  its  bounding  surface  varies  directly  as  the 
distance  from  the  centre  of  the  sphere  (Art.  1338).  If  the 
earth  were  such  a  homogeueoua  sphere,  and  if  AOA'  (Pig. 
80)  represented  a  shaft  running  straight  through  its  centre 
from  surfitce  to  surface,  then,  if  a  particle  were  free  at  one 
end.  A,  it  would  move  to  the  centre  of  the  earth,  0,  where 
its  velocity  would  be  a  maximum,  and  thence  on  to  the 
opposite  side  of  the  earth.  A',  where  it  would  come  to  rest ; 
then  it  would  return  through  the  centre,  O,  to  the  j-  ie.  A, 
from  where  it  suirted  ;  and  its  motion  would  continue  to  bo 
oscillatory,  and  thus  it  would  move  backwards  and  forwards 
from  one  side  of  the  earth's  surface  to  the  other,  and  the 
time  of  the  oscillation  would  bo  independent  of  the  earth's 
radius;  that  is,  at  whatever  point  within  the  earth's  sur&ce 
the  particle  be  placed  it  would  reach  the  centre  in  the 
same  time. 


Sit 


■  t     ' 


298 


A    VARtABhM  BSPITLBIVK  t^jXOX. 


in 


jjifr 


i 


OoB. — To  find  this  time.    Since  /i  is  the  attraction  at  a 
unit  of  distance  and  g  the  attraction  at  the  distance  B^  we 


hare  fi  =  -^,  which  in  t  =.  — j  gives 


for  the  time  it  wonld  take  a  body  to  moTe  from  any  point 
within  the  earth's  surface  to  the  centre. 
If  we  put  g  =  Z^  feet  and  R  =  3963  miles  we  get 

^  =  21  m.  6  s.  about, 

which  would  be  the  time  occupied  in  passing  to  the  earth's 
centre,  however  near  to  it  the  body  might  be  placed,  or 
however  far,  so  long  as  it  is  within  th<)  surface. 

172.  Motton  of  a  Particle  under  the  Action  of  a 
Variable  RspnlaiTe  Force. — Let  the  force  be  one  of 
repulsion  and  vary  as  the  distance,  then  the  equation  of 
motion  is 

dhi  ~ 


dt» 


=  fix. 


Let  us  suppose  the  particle  to  be  projected  from  the  cen- 
tre of  force  with  the  velocity  v^  ;  then  we  have 


dt* 


F«^  +  »,» ; 


(1) 


As  t  increases  x  also  increases,  and  the  particle  recedes 
further  and  further  from  the  centre  of  force;  and  the 
velocity  also  increases,  and  ultimately  equals  ao  when  x  =r 
t  =  cc.    Thus  in  this  case  the  motion  is  not  oscillatory. 


ii 


ffjRCB. 

9  the  attraction  at  a 
b  the  distance  R^  we 


OTe  from  any  point 
3  miles  we  get 


wiing  to  the  earth'g 
aight  be  placed,  or 
surface. 

r  the  Aetioii  of  a 

he  force  be  one  of 
en  the  equation  of 


ected  from  the  oen- 
ire  have 


(1) 


*'> 


the  particle  recedes 
of  force;  and  the 
equals  00  when  x  =r 
is  not  oscillatory. 


'■imm 


ivmtimmmmm^jii  '-'Mi    i  iiiiiiiiiii|| 


•mm 


A   VASTABIX  ATTHACnva  POSOX. 


390 


X73.  Motton  of  «  Particle  under  the  Aotioa  of  an 
Attractive  Force  which  ia  in  the  line  of  motion,  and 
which  variea  Invervely  aa  the  Square  of  the  Uatance 
from  the  Centre  of  Force. 

Let  O  (Fig.  80)  be  the  centre  of  force,  P  the  position  of 
the  particle  at  the  time  t;  and  A  the  position  at  rest  when 
<  =  0,  80  that  the  particle  starts  from  A  and  moves  to- 
wards 0.  Let  OP  =  X,  Ok  =  a,  and  /«  =  the  absolute 
force  as  before  or  the  acceleration  at  unit  distance  firom  0. 
Then  the  equation  of  motion  is 


ft 


dx 


Multiplying  by  2  tj  and  integrating,  we  get 


dt* 


=^e-^. 


(1) 


which  gives  the  velocity  of  the  particle  at  any  distance,  x, 
from  the  origin. 
From  (1)  we  have 


2/u\/< 


ax 


dt 


the  negative  sign  being  taken  because  in  the  motion  to- 
wards 0,  X  diminishes  as  /  increases.    This  gives 


^J^dt^ 


—  xdx 


=  u 


[' 


a  —  2x 


Vox  —  afl      2  y/ax  _  ^^  J 


\dx. 


■'*  # 


800 


VMLOCtrr  IN  FALLING. 


Integrating  and  taking  the  lunits  corresponding  to  ^  s  ^ 
and  /  =  0,  we  have 


(2) 


which  gives  the  valne  of  t 

When  the  particle  arrives  at  0,  a;  =:  0,  therefore  the 
time  of  felling  to  the  centre  0  from  A  is 


From  (1)  we  see  that  the  velocity  =  0  when  a;  =  a ;  and 
=  «  when  a;  =  0 ;  hence  the  velocity  increases  as  the 
particle  approaches  the  <»ntre  of  forco,  and  ultimately, 
when  it  arrives  at  the  centre.,  becomes  infinite.  And 
although  at  any  point  very  near  to  0  there  is  a  very  great 
attraction  tending  towards  0,  at  the  point  0  itself  there  is 
no  attraction  at  all;  therefore  the  particle,  approaching 
the  centre  with  an  indefinitely  great  velocit;y,  must  pass 
through  it.  Also,  everything  being  the  same  at  equal 
distances  on  either  side  of  the  centre,  we  see  that  the 
motion  must  be  retarded  as  rapidly  as  it  was  accelerated, 
and  therefore  the  particle  will  proceed  to  a  point  A'  at  a 
distance  on  the  other  side  of  0  equal  to  that  from  which  it 
started  ;  and  the  motion  will  continue  oscillatory. 

174.  Vdloeiiy  acquired  in  Falling  throng  a  Oraat 
Height  above  the  Earth.— The  preceding  case  of  motion 
includes  that  of  a  body  falUng  iVom  a  great  height  above 
the  earth's  surface  towards  its  centre,  the  distance  through 
which  it  falls  being  so  great  that  the  variations  of  the  earth's 
attraction  due  to  the  distance  must  be  taken  into  account 
For  a  sphere  attracts  an  external  particle  with  a  force  which 
varies  inversely  as  the  square  of  the  distance  of  the  particle 


a. 


■espondrng  tot  x  t 


1+™]    (') 


=  0,  therefore  the 


)  when  a;  =  a ;  and 
Y  increases  as  the 
0,  and  ultimately, 
kes  infinite.  And 
lere  is  a  very  great 
nt  O  itself  there  is 
rticle,  approaching 
relocit;y,  must  pass 
;he  same  at  equal 
,  we  see  that  the 
1  it  was  accelerated, 
to  a  point  A'  at  a 
that  from  which  it 
scillatory. 

;  tliroii^  a  Or««t 
ling  case  of  motion 
great  height  above 
B  distance  through 
ations  of  the  earth's 
taken  into  account 
)  with  a  force  which 
iuce  of  the  particle 


fit  il.lWIHH 


ivapii 


vtLocirr  m  fallino. 


301 


from  the  centre  of  the  sphere  (Ait  133a);  therefore  if  i?  is 
the  earth's  radius^  g  the  kinetic  measure  of  gravity  on  a 
unit  of  mass  at  the  earth's  surface  (Arts.  20,  23),  and  z  the 
distance  of  a  body  from  the  centre  of  the  earth  at  the  time 
t,  then  the  equation  of  motion  is  • 


if  ^' 


which  is  the  same  as  the  equation  in  Art  173  by  writing  n 
for  gR" ,  therefore  the  results  of  the  last  Art.  will  apply  to 
this  case.    Substitu'ang  gIP  for  /*  in  (1)  of  Art  173  we 

have 

/«  --  x\  ^y 


«"  =  ^*(~> 


When  the  body  reaches  the  earth's  surfoce,  x  =  R  and 
(1)  becomes 

^^%gR(^^).  P) 

H  o  is  infinite  (3)  becomes 

so  that  the  velocity  can  never  be  so  great  as  this,  however 
fiir  the  body  may  fall;  and  hence  if  it  were  possible  to 
project  a  body  vertically  upwards  with  this  velocity  it  would 
go  on  to  infinity  and  never  stop,  supposing,  of  course,  that 
there  is  no  resisting  medium  nor  other  disturbing  force. 
If  in  (2)  we  put  g  =  32^  feet  and  R  =  3963  miles  we 

get  , 

V  =  [2- 321-3963. 6280]* feet  =  6-95  miles; 

BO  that  the  greatest  possible  velocity  which  a  body  (»n 
acquire  in  falling  to  the  earth  is  less  than  7  miles  per 
second,  and  if  a  body  were  projected  upwards  with  that 


"■^s^^as^agyAjEFgCi^'S 


mm^mmmm 


303 


MOTION  IN  A  RXaiSTlNQ  MSDIUJl. 


I*; 


veloci  V,  and   were   to  meet  with    no  resistance  except 
gravity,  it  would  never  return  to  the  earth. 

Cob.— To  find  the  velocity  which  a  body  would  acquire 
in  falling  to  the  earth's  surface  from  a  height  h  above  the 
surface,  we  have  from  (1)  by  putting  z  =  Rm^a  =  h  +  R, 

If  A  be  small  com^)ared  with  Ji,  this  may  be  written 


which  agrees  with  (6)  of  Art  140. 

The  laws  of  force,  enumerated  in  Arts.  171,  173,  are  the 
only  laws  that  are  known  to  exist  in  the  universe  (Pratt's 
Mecbs.,  p.  212). 

175.  Motion  in  a  Resiatisg  Modiun.— In  the  pre- 
ceding discussion  no  account  is  taken  of  the  atmospheric 
resistance.  We  shall  now  consider  the  motion  of  a  body 
near  the  surfoce  of  the  earth,  taking  into  account  the 
resistance  of  the  air,  which  we  may  assume  varies  as  the 
square  of  the  velocity. 

A  particle  under  the  action  of  gravity,  as  a  constant  force, 
moves  in  the  air  supposed  to  be  a  resisting  medium  of 
uniform  density,  of  which  the  resistant^  varies  as  the  square 
of  the  velocity  required  to  determine  the  motion. 

Suppose  the  particle  to  descend  towards  the  earth  from 
rest  Take  the  origin  at  the  starting  point,  let  the  line  of 
its  motion  be  the  axis  of  x ;  and  let  x  be  the  distance  of 
the  particle  from  the  origiii  at  the  time  t,'  and  for  con- 
venience let  gifi  be  the  resistance  of  the  air  on  the  particle 
for  a  unit  of  velocity;  gJi^  is  called  the  eoeffioient  of  resist- 
ance.   Then  the  resistance  of  the  air  at  the  distance  x  fh>m 


rtte 


MBDIUM. 

0  resistance  except 
rth. 

body  would  acquire 

height  ft  above  the 

=  iff  and  a  =  A  +  i?, 

fiffRh 
R  +  h' 

Y  be  written 


».  171,  173,  are  the 
le  UDiTerse  (Pratt's 


iiun. — ^In  the  pre- 

)f  the  atmospheric 

motion  of  a  body 

into  account  the 

lume  Taries  as  the 

M  a  constant  force, 
fisting  medium  of 
'aries  as  the  square 
otion. 

is  the  earth  from 
nt,  let  the  line  of 
be  the  distance  of 
e  t,'  and  for  con- 
^ron  the  particle 
^ffii^ient  of  resist- 
te  distance  x  fh>m 


>>«;W«V«<bwr^tMT«i4«fri«m9a^ian:4aaet*vMi^  • 


MOTION  Ilf  A  RJHSISmrO  MJSDtUM. 

the  origin  is  gi^  (^) ,  which  acts  upwards,  and  the  force  of 
gravity  is  g  acting  downwards,  the  mass  l)eing  a  unit. 
Hence  the  equation  of  motion  is 


(1) 


d 


.♦.    gdt  = 


m 


-Q' 


Integrating,  remembering  that  when  ^  =  0,  t>  =  0,  we 


get 


1  +  h- 
t  =  ^log ^,  (Calculus,  p.  259,  Ex.  5). 


2k 


^      "dt 


Passing  to  exponentials  we  have 


dt 


ke^g*  +  e-^*' 


(«) 


which  gives  the  velocity  in  terms  of  the  time.    To  fina  it  in 
terms  of  the  space,  we  hav«  from  (1) 


««(§)' 


-^=:2gi^dx; 


observing  the  proper  limits; 


m 


if 


L 

fit 


■■-'WfiiiifflWifflgitl^^ 


3C4 


MOTION  or  ASCENT  IN  TETM  AlB, 


which  gives  the  velocity  in  terms  of  the  distance. 


(4) 


Also,  integrating  (2)  taking  the  same  limits  as  before, 
we  get 

gli^x  =  log  (e*»*  +  C"***)  —  log  2  ; 


2«ff»'«  =  «*»«  +  e-*Ot, 


(5) 


which  gives  the  relation  between  the  distance  and  the  time 
of  falling  through  it. 

As  the  time  increases  the  term  e-*v  diminishes  and  from 
(5)  the  space  increases,  becoming  infinite  when  the  time  is 
infinite;  bat  from  (2),  as  the  time  increases  the  velocity 
becomes  more    nearly  uniform,   and  when    <  =  oo,   the 

velocity  =  r ;  and  although  this  state  is  never  reached,  yet 

it  is  that  to  which  the  motiuu  approaches. 

176.  Motion  of  a  Particle  Aaeending  in  the  Air 
agsdnst  tbe  Action  of  Oravity.— Let  ns  suppose  the 
particle  to  be  projected  upwards,  that  is,  in  a  direction 
contrary  to  that  of  the  action  of  gravity,  with  a  given 
velocity,  v,  it  is  required  to  detennine  the  motion. 

Let  UB  suppose  the  particle  to  be  of  the  same  form  and 
siso  as  before,  and  the  same  coefficient  of  resistance. 
Then,  taking  x  positive  upwards,  both  gravity  and  the 
resistance  of  the  air  tend  to  diminish  the  velocity  as  t 
increases;  so  tliat  the  equation  of  motion  is 


tPx 


-,<)•; 


(1) 


(4) 

istance. 
limits  as  before, 

log  2; 

(5) 
mce  and  the  time 


linishes  and  from 
when  the  time  is 
aases  the  velocity 
len    ^  =  00,   the 

never  reached,  yet 


ling  in  the  Air 

as  suppose  the 
8,  in  a  direction 
ty,  with  a  given 
motion. 

le  same  form  and 
it  of  resistance, 
gravity  and  the 
the  velocity  as  t 
is 


(1) 


MOTION  or  ASCMXT  IX  TBM  .AIM, 
dx 


dh 


dl 


i  +  KI) 


.'.    tan->  1e^  —  *»«"*  (H  -  O^d » 

(Calculus,  p.  244,  Ex.  3),  since  the  initial  velocity  is  v. 

Taking  the  tangent  of  both  members  and  solving  for 

dx  . 

^  _  1      vl  —  tan  kgt  . 
di  ~  k'  1  +  vk  toD  kgt* 


(8) 


which  gives  the  velocity  in  terms  of  the  time.    To  find  it 
in  terms  of  the  difltanoe,  w»  1mv(>  from  (1) 


-(f)' 


-,  s=  -«jfifc»<toi 


m 


...    log     ^^^    =:^%gm', 

.. .  (^)'  =  ^-*^'^  -  ^  (1  -  ^**'''>' 

which  gives  the  velocity  in  terms  of  the  distance. 

Also,  integrating  («)  irfter  sabstituting  sine  and  cosine 
for  tangent,  and  taking  the  same  limits  w  before,  we  get 

^/L*»  =3  log  (»*  sin /is''  +•  oo*  M) ;  (*) 

which  gives  the  space  desori'jed  by  the  particle  in  terms  of 
tiie  tim«. 


i 


UBI 


mozaon  or  AsoMifT  isr  tsjb  air. 


OoB.  1.— To  find  the  greatest  height  to  which  the  par- 
tide  will  ascend  put  the  velocity,  -^  =  0,  in  (3)  and  get 


which  is  the  distance  of  the  highest  point 

da 
Putting  ^  =  v>  in  .^s    Te  ge*; 


(«) 


kg 


m 


which  is  the  time  required  for  the  particle  to  reach  the 
highest  point.  TIaving  reached  the  greatest  height,  the 
particle  will  begin  to  foil,  and  the  circnmfitances  of  the 
fall  will  be  giyen  by  the  equations  of  Art  175. 

OoB.  2.— Since  h  is  the  same  in  this  and  Art  176,  we 
may  compare  the  velocity  of  projection,  t',  with  that  which 
the  parcicle  would  acquire  in  descending  to  the  point 
wh<  nee  it  was  projected.  Denote  by  v^  the  velocity  of 
the  particle  when  it  reaches  the  '^nint  of  starting.  From 
(3)  of  Art  176  we  have 


and  placing  this  value  of  x  equal  to  that  given  in  (6), 
we  get, 


r-w:*^^-^^' 


.'.    »,  = 


(l  +  *»l^)*' 

which  is  less  than  t>;  hence  the  velocity  »^aired  in  the 


•Ai 


f.W"WtW»K5?2?l 


m 


S  AIR, 


t  to  which  the  par- 
0,  in  (3)  and  get 


(6) 


t 


(7) 

irtiole  to  reach  the 
latest  height,  the 
cnm^tances  of  the 
;.  175. 

and  Art  176,  we 
t>,  with  that  which 
ling  to  the  point 
t'o  the  velocity  of 
)f  starting.    From 


that  given  in  (6), 


ty  »^aired  in  the 


wmmmm 


UOTIOH  OF  A  PSOJBCTtLB, 


307 


descent  is  less  than  that  lost  in  the  ascent,  as  might  have 
been  inferred. 

Cob.  3.— Substituting  (6)  in  (5)  of  Art  176,  we  get  for 
the  time  of  the  descent. 


/ 


^log(Vl  +  iV  +  *»), 


which  is  difFennt  from  the  time  of  the  ascent  as  given  in 
(7).  (See  Pri  je's  Anal.  Mech's,  Vol.  I,  p.  406 ;  Venturoli's 
Mech's,  p.  81) ;  Tait  and  Steele's  Dynamics  of  a  Particle, 


■AHi. ) 


177.  Motion  of  a  Protjectile  in  •  Resisting  M^- 
dimn. — The  theory  of  the  motion  of  projectiles  in  vacuo, 
which  waf;  examined  under  the  head  of  Kinematics,  affords 
results  which  differ  greatly  from  those  obtained  by  direct 
experiment  in  the  atmosphere.  When  projectiles  move 
with  but  small  velocity,  the  discrepancy  between  the  para- 
bolic theory,  and  what  is  found  to  occur  in  practice,  is 
small ;  but  with  increasing  velocities,  as  those  with  which 
bftils  and  shells  traverse  their  paths,  the  air's  resistance 
increases  in  a  higher  ratio  than  the  velocity,  so  that  the 
discrepancy  becomes  very  great. 

The  most  important  application  of  the  theory  of  projec- 
tiles>  is  that  of  Gunnery,  in  which  the  motion  takes  place 
in  the  air.  If  it  were  allowable  to  negWct  the  resistance  of 
the  air  the  investigations  in  Part  II  would  explain  the 
theory  of  gunnery ;  but  when  the  velocity  is  considerable, 
the  atmospheric  resistance  changes  the  nature  of  the  tra- 
jectory so  much  as  to  render  the  conclusions  drawn  from 
the  theory  of  projectiles  in  vacuo  almost  entirely  inap- 
plicable in  practice. 

The  problem  of  gunnery  may  be  stated  as  follow^: 
Given  a  projectile  of  known  weight  and  dimensions, 
starting  with  a  known  velocity  a^  a  known  angle  of  eleva- 


308 


XOTtON  or  A  PKOJBCTTLE. 


tion  in  a  calm  atmosphere  of  approximately  known  density ; 
to  find  its  range,  time  of  flight,  velocity,  direction,  and 
position,  at  any  moment ;  or,  in  otb'jr  words,  to  construct 
its  trajectory.  This  problem  is  not  yet,  however,  suscepti- 
ble of  rigorous  treatment ;  mathematics  has  hitherto  proved 
unable  to  furnish  complete  formulae  satisfying  the  condi- 
tions. The  resistance  of  the  air  to  slow  movements,  say  of 
10  feet  per  second,  seems  to  vary  with  the  first  power  of 
the  velocity.  Above  this  the  ratio  increases,  and  as  in  the 
case  of  the  wind,  is  usually  reckoned  to  vary  as  the  square 
of  the  velocity ;  beyOnd  this  it  increases  still  further,  till  at 
1200  feet  per  second  the  resistance  is  found  to  vary  as  the 
cube  of  the  velocity.  The  ratio  of  increase  after  this  point 
is  passed  is  supposed  to  diminish  again ;  but  thoroughly 
satisfactory  data  for  its  determination  do  not  exist. 

From  experiments*  made  to  determine  the  motion  of 
cannon-balls,  it  appears  that  when  the  initial  velocity  is 
considerable,  the  resistance  of  the  air  is  more  than  20  times 
as  great  as  the  weight  of  the  ball,  and  the  horizontal  range 
is  often  a  small  fraction  of  that  which  the  theory  of  pro- 
jectiles in  vacuo  gives,  so  that  the  form  of  the  tngectory  is 
very  different  firom  that  of  a  parabolic  path.  Such  experi- 
ments have  been  made  with  great  care,  and  show  how  little 
the  parabolic  theory  is  to  be  depended  upon  in  determining 
the  motion  of  military  projectiles. 


178.  Motion  of  «  Prc^otile  in  the  Atmosphere 
Snppoeing  its  Resistance  to  wary  ss  the  Square  of 
the  Telocity. — A  particle  under  the  action  of  gravity  ia 
projected  from  a  given  point  in  a  given  direction  loith  a 
given  velocity,  and  moves  in  the  atmosphere  whose  resistance 
is  assumed  to  vary  as  the  square  of  the  velocity ;  to  deter- 
mine  the  motion. 


*  B«e  BDcrolo)Midto  BrllMiiiiea,  Art  ChuuMrr ; 
Hnttoa'f  Tncu. 


•iw  Bobln'i  Qnaasrj,  and 


IJP. 

3ly  known  density; 
ity,  direction,  and 
ords,  to  construct 
however,  suscepti- 
las  hitherto  proved 
isfying  the  condi- 
movements,  say  of 
the  first  power  of 
ses,  and  as  in  the 
rary  as  the  square 
till  farther,  till  at 
nd  to  vary  as  the 
se  after  this  point 
;  but  thoroughly 
not  exist, 
le  the  motion  of 
initial  velocity  ia 
lore  than  20  times 
;  horizontal  range 
the  theory  of  pro- 
I  the  trajectory  is 
th.  Such  experi- 
d  show  how  little 
m  in  determining 


M  Atmoaphera 
■  the  Square  of 

fion  of  gravity  ia 

direction  with  a 

a  whoae  reaiatanca 

'eiocitjf  ;  to  deter- 


Bobln'i  OnniierT,  aaA 


mwaOH  OP  A  PROJEOfiLA 

Take  the  given  point  as  origin,  the  axis  of  x  horizontal, 
the  axis  of  y  vertical  and  positive  upwards,  so  that  the 
direction  of  projection  may  be  in  the  plane  of  xy.  Let  v 
be  the  velocity  of  projection,  g  the  acceleration  of  gravity, 
a  the  angle  between  the  axis  of  x  and  the  ^'le  of  projection, 
and  let  the  resistance  of  the  air  on  the  particle  be  i  for  a 
nnit  of  velocity ;  then  the  resistance,  at  any  time,  t,  in  the 

line  of  motion,  is  h  ijA  ;  and  the  x-  and  y-components  of 

thie  resistance  art,  respectively, 


,  da    dx         J 
*  jj  •  TV,    and 
dt    dt 


da    dy 
dt    dt 


Then  the  equationfl  of  motion  are,  resolving  horizontally 

and  vertically, 

(Pa;  _       ,da    dx  .^v 

dfl-  ~^dt  rr  ^  ' 


dP  ~ 


9 


Prom  (1)  we  have 
tdx\ 


dx 
dt 


=  -h-la; 


dx 


log 


^dt   dt 

dm 
dt 

voosa 

(») 


—  ha\ 


since  when  <  =  0,  -^  =:^  v  c(x  a', 


dx 
di 


=  v  cos  «  «-■ 


(8) 


Maltildying  (1)  and  (2)  by  dy  and  dx,  respectively,  and 
aabtraoting  the  former  from  the  latter  we  have 


d^dx  —  iPxdy  _  ^ 


dfi 


gdx. 


(4) 


ii 


310  MonoN  or  a  projbctilm. 

Substituting  in  (4)  for  dP  ita  value  from  (3)  we  get 

Substituting  in  the  second  member  of  (6)  for  dx  its  value 
dxV  ^  da«/ i;» co8»  - '^'"* 


(6) 


Pat  ^  =  P'  and  (6)  becomes 


"+'■>**  =  pi?-.  •**• 

Integrating,  and  remembering  that  when  «  =  0,^  =  tan  <^ 
we  get 

P  (1  +  J»«)*  +  log  [p  +  (l+  JB»)*] 


-^,-e« 


(7) 


Ar*  Cf /8»  a 
where  e  is  the  constant  of  integration  whose  value 

=  tan  o  sec  «  +  log  (tan  o  +  sec  «)  +  ,     -^  .    .   fg^ 

'        «t)»COS»o     ^  ' 

From  (5)  we  have 

t>»oos»a  dxXdxf' 

which  in  (7)  gives 

i'a  +i^)*  +  log  [i»  +  (1  +i,»)*]  -  c  =  ^|, 
, dp 

"     /'(l+jJ^+l0g[iB  +  (l+^)*]_c^**''       ^*^ 


ttHH 


tM. 

(3) 

we  get 

9 

% 

(6) 

i)  for  dx  its  value 

«»»»(fo. 


(6) 


»rf». 


«  =  0,jp  =  tan 


«> 


/^)*] 


Mseralne 


ifct)»C08»tt 


(7) 


(8) 


—  =  kdx,     (9) 


.uu  l.i'.JCJiWgWWWff^^^^Wff!— W^gl'^ff'wW 


and 


MOTtON  or  A  PROJMCTILa. 

pdp 


311 
=:Hy.   (10) 


/» (1  +i^*  +  log  O  +  (1  +  pO*]  -  c 
From  (4)  ..e  have 

dX'dp  =  —  gdfi. 

Sabgtituting  this  value  of  dx  in  (9)  and  solving  for  dt  we 
get 

|c  -  p  (1  +  p»)*  ~  log  [/»  +  (1+P»)*]  f* 

the  negative  sign  of  dp  being  taken  because  j»  is  a  decreas- 
ing function  of  ^. 
Bephicing  the  value  of  i>  =  ^,  (9),  (10),  and  (U)  become 


(fo  =    T 


4^ 

dx 


*i(i^^Wwl^(^*]-«' 


,     1 


dx    dx 


'l('+g)*-'-[|+('  +  g)*]-«' 


-d 


dt=: 


dx 


♦^{-K'+gr-Hi^^OT" 


(A) 


(B) 


,  (0)- 


from  which  equations,  were  it  possible  to  integrate  them, 
X,  y,  and  t  might  be  found  in  terms  o*  ^  >  *»^  *'  ^  ""^^ 

eliminated  from  the  two  intends,  of  (A)  and  (B),  the  re- 
sulting equation  in  terms  of  x  and  y  would  be  that  of  the 


I 
1 


1 1 


312 


Monoir  or  a  pxwjwmjk 


required  trajectory.  But  these  eqnatiors  cannot  be  inte- 
grated in  6nit6  terms;  only  approximate  .eolations  of  them 
can  be  made  ;  and  by  means  of  these  the  path  of  the  pro- 
jectile may  be  constructed  approximately.  (See  VentoroM's 
Mechs.,  p.  92.) 

Squaring  (A)  and  (B),  and  dividing  their  suit  by  the 
square  of  (0)  we  get 


di* 


9 
k 


1+^ 


-n'^%r-^<^^\^ 


+  V 


^m 


(D) 


which  gives  the  velocity  in  terms  of  ^. 


179.  Motion  of  a  Projectile  in  the  Atmosphere 
under  a  small  Angle  of  Elevation.— The  case  fre- 
quently occurs  in  phwtice  where  the  angle  of  projection  }* 
very  small,  and  where  the  projectile  rises  but  a  very  little 
above  the  horizontal  line.  In  this  case  the  equation  of  the 
part  of  the  trajectory  that  lies  above  the  horizontal  line 
may  easily  bo  found;  for,  the  angle  of  projection  being 

dy 
very  small,  ^  will  be  very  small,  and  fterefore,  throughout 

the  path  on  the  upper  side  of  the  axis  of  x,  powers  of 

^  higher  than  the  first  may  be  neglected.    In  this  case 

then 

diszix',       .•.    »  =  «} 


which  in  (6)  of  Art.  178,  becomes 


rfy 


d^  = 


«»co«*« 


«**<fe; 


»r8  cannot  be  inte- 
«  solutions  of  them 
he  path  of  the  prp- 
j.    (Sm  Ventiuroli's 

;  their  sun?  by  the 


(D) 


/'  _.  ^\* 


\* 


the  Atmosphere 

•n. — The  case  fre- 
{le  of  projection  }# 
ses  bat  a  very  little 
he  equation  of  the 
he  horizontal  line 
f  projection  being 

irefore,  throughoat 

cis  of  X,  powers  of 

ited.    In  this  case 


iiiTjji.iiiimfjwii.il 


BXAMFLBa. 


913 


Integrating,  we  get 

tana  =  — 


dx 


2kir  cos'  a 


(«*»-!); 


dv 
since  when  x  =  0,  -^  =:  iaa  a. 


Integrating  again  we  get 


y  =2;tan  «  -f 


«      4ifct;»  cos*  o  ^  /    \  / 


2;{;t;»  COB* 

Expanding  e**^  in  a  series,  (1)  becomes 

_  ga^  gkafi 

y  _  a:  tan  «  -  c^^^^  "  3«»  cos*  «  ~ 


(2) 


the  first  two  terms  of  wHich  represent  the  i  ijectory  in 
vacno.     [See  (3)  of  Art.  151.] 
From  (3)  of  Art.  178,  we  have 


dt  = 


dx 


t  - 


vcosa 
flto-1 


V  cos  a 


m 


which  gives  the  time  of  flight  in  terms  of  the  absdraa. 

The  most  complete  and  valuable  series  of  experiments 
on  the  motion  of  projectiles  in  the  atmosphere  that  has  yet 
been  made,  is  that  of  Prof.  F.  Bashforth  at  Woolwich. 


EXAMPLES. 


1.  Find  how  far  a  force  equal  to  the  weight  of  n  Ibe., 
would  move  a  weight  of  m  lbs.  in  /  seconds ;  and  find  the 
velocity  acqaiied. 


Here  P  =  n,  and  W=zm;  therefore  from  (1)  of  Art 
"^  we  have 


**  ==  — /; 


•  •   /  —  — > 


which   in  (4)  and  (5)  respectively  of    (Art.    10),  gives 


tn 


;  and  «  =  4  -^  /*. 
*  m 


2.  A  body  weighing  n  lbs.  is  moved  by  a  constant  foroe 
which  generates  in  the  body  in  one  second  a  velocity  of  a 
feet  per  second ;  find  the  force  in  pounds. 


Ana. 


5?  lbs. 
9 


3.  Find  in  what  time  a  force  of  4  lbs.  wonld  move  a 
weight  of  9  lbs.  through  49  ft  along  a  smooth  horizontal 
plane ;  and  find  the  velocity  acquired. 

21 

4.  Find  the  number  of  inches  through  which  a  force  of 
one  ounce,  constantly  exerted,  wiU  move  a  mass  weighing 
one  lb.  in  half  a  second.  ^„,,  3^  /ijs, 

6.  Two  weights,  P  and  Q,  are  connected  by  a  string 
which  passes  over  a  smooth  peg  or  pulley ;  required  to 
determine  the  motion. 

Since  the  peg  or  pulley  is  perfectly 
smooth  the  tension  of  the  string  is  the 
same  throughout;  hence  the  foroe  which 
causes  the  motion  is  the  difference  between 
the  weights,  P  and  Q,  the  weight  of  the 
string  being  neglected.  The  moving  force 
therefore  is  P  —  0;  but  the  weight  of  the 
mass  moved  is  P  +  ^.  Hence  substituting 
in  (1)  of  Art  25,  we  get 

P+Q 
9 


X 


Q* 


■p-e 


'f. 


Fig.  SOo. 


•re  from  (1)  of  Art 


m 


i    (Art.    10),  gives 


by  a  constant  force 
Jond  a  velocity  of  a 


is. 


Ana.  —  lbs. 


lbs.  would  move  a 
emooth  horizontal 


21 


V^' 


V  =  ^t. 


gh  which  a  force  of 
e  a  mass  weighing 
Am.  3ffH)\ 

leoted  by  a  string 
alley ;  required  to 


Btly 
the 
lich 
een 
the 
>rce 
the 
ing 


X 


Q* 


Fig.  SOo. 


•mtt'mi'twmvtagmmmw'imimiPifsmrmiUKa 


XXAMPLSa. 


.-.  /= 


P-Q. 


which  is  the  acceleration. 
Substituting  this  in  (4)  and  (5)  of  Art  10,  we  have 


815 


(1) 


V  =  -i^.— j<//, 


P—  o 


(8) 
(8) 


which  giTCS  the  velocity  and  space  at  the  time  t,  the  initial 
velocity  v,  being  0. 

6.  A  body  whose  weight  is  Q,  rests  on  a  smooth  hori- 
zontal table  and  is  drawn  along  by  a  weight  P  attached  to 
it  by  a  string  passing  over  a  pulley  at  the  edge  of  the  table ; 
find  the  motion  of  the  bodies. 

Since  the  weight  Q  is  entirely  supported  by  the  resistance 
of  the  table,  the  moving  force  is  the  weight  P,  hanging 
vertically  downwards,  and  the  weight  of  the  mass  moved  is 
P  +  Q;  therefore  from  (1)  we  have 


/  = 


P  +  Q 


(I) 


and  this  in  (4)  and  (5)  of  Art.  10  gives  the  velocity  and 
space. 

7.  Required  the  tension,  T,  of  the  string  in  the  pre- 
ceding example. 

Here  the  tension  is  evidently  that  force  which,  acting 
along  the  string  on  the  body  whose  weight  is  Q,  produces 

p 
in  it  the  acceleration,  -„  ,  ^g,  and  therefore  is  measui^ 

P  +  Y 
by  the  mass  of  Q  into  its  acceleration.    Henoe 


m 


316 


JfXAMPLSa. 


'8.  Find  the  tension,  T,  of  the  string  in  Ex.  5. 
Here  the  tension  equals  the  weight  Q,  pins  the  force 
which,  acting  along  the  string  on  Q,  produces  in  it  the 
acceleration 

P+Q^' 

••    ^-^^-g'PTQ^' 

-    ^^g 

~  P  +  Q' 

or  it  equals  P  the  accelerating  force  which,  of  course, 

gives  the  same  result, 

9.  Two  weights  of  9  lbs.  and  7  lbs.  hang  over  a  pulley,  as 
in  Ex.  5 ;  motion  continues  for  5  sees.,  when  the  string 
breaks ;  find  the  height  to  which  the  lighter  weight  will 
rise  after  the  breakage. 

Substituting  in  (2)  of  Ex.  5  we  have 

v  =  V^32.5  =  20; 

therefore  each  weight  has  a  velocity  of  20  feet,  when  the 
string  breaks.  Hence  from  (6)  of  Art.  10,  we  have  (calling 
^32ft.) 

that  is,  the  lighter  weight  will  rise  6J  feet  before  it  begins 
to  descend. 

10.  A  steam  engine  is  moving  on  a  horizontal  plane  at 
the  rate  of  30  miles  an  hour  when  the  steam  is  turned  ofE ; 
supposing  the  resistance  of  friction  to  be  ^  of  the  weight, 
find  how  long  and  how  far  the  engine  will  run  before  it 
stops. 


PQ 

:  in  Ex.  5. 

;  Q,  pins  the  force 
produces  in  it  the 


■^9, 


irce  which,  of  course, 

uig  over  a  pulley,  as 

58.,  when  the  string 

lighter  weight  will 


>f  20  feet,  when  the 
10,  we  have  (calling 


feet  before  it  begins 

horizontal  plane  at 
team  is  turned  off ; 
e  jfy  of  the  weight, 
le  will  run  before  it 


BXAMPhK3. 


»17 


Let  ff  be  the  weight  of  the  engine;  then  the  rosistance 

W 
of  friction  is  -r^t  and  tl»8  is  directly  opposed  to  motion, 
400 


w 

m 


9^' 


400 
30x1760x8 


=:44 


The  velocity,  v,  is  30  miles  an  hour  =  — ^  ^  g^ 

feet  per  second.    Substituting  these  Vfiues  of  /  and  v  in  the 
equation  v  =  //,  we  get 

44  =  ^«; 

,'.    t  =  550  sees., 

which  is  the  time  it  will  take  to  bring  the  engine  to  rest  if 
the  velocity  be  retarded  ^  feet  per  second. 
Also  r»  =  Ufa,  therefore 

g  —  ««yy^x*oo  —  12100  feet. 

11.  A  man  whose  weight  is  W,  stands  on  the  platform 
of  an  elevator,  as  it  descends  a  vertical  shaft  with  a  uniform 
acceleration  of  |jf ;  find  the  pressure  of  the  man  upon  the 
platform. 

Let  P  be  the  pressure  of  the  man  on  the  platform  when 
it  is  moving  with  an  acceleration  of  \g ;  then  the  moving 
force  is  W  —  P;  and  the  weight  moved  is  W',  therefore 


W 
W^P  =  ^\9; 


tr. 


J2.  A  plane  supporting  a  weight  of  12  ozs.  is  descending 
with  a  uniform  acceleration  of  10  ft.  per  second ;  find  ,the 
pressure  that  the  weight  exerts  on  the  plane. 

Am.  S\  ozs. 


318 


SXAMPLSa. 


13.  A  weight  of  24  lbs.  hanging  over  the  edge  of  a 
smooth  table  drags  a  weight  of  12  lbs.  along  the  table; 
find  (1)  the  acceleration,  and  (2)  the  tension  of  the  string. 
Aw.  (1)  5^  ft.  per  sec. ;  (2)  20  lbs. 

||;  !*•  A  weight  of    8  lbs.    rests    on    a    platform;    find 

its  pressure  on  the  platform  (1)  jf  the  latter  is  de- 
scending with  an  acceleration  of  \g,  and  (2)  if  it  is 
ascending  with  the  same  acceleration. 

Ans.  (1)  7  lbs.;  (2)  9  lbs. 

15.  Two  weights  of  80  and  70  lbs.  hang  over  a  smooth 
pulley  as  in  Ex.  5  ;  find  the  space  through  which  they  will 
move  from  rest  in  3  sees.  A7i8.  9|  ft. 

16.  Two  weights  of  16  and  17  ounces  respectively  hang 
over  a  smooth  pulley  as  in  Ex.  5 ;  find  the  space  de- 
scribed and  the  velocity  acquired  in  five  seconds  from  rest 

Ana.  »  =  26,  V  =  10. 

17.  Two  weights  of  5  lbs.  and  4  lbs.  togethei  pull  one 
of  7  lbs.  over  a  smooth  fixed  pulley,  by  means  of  a  con- 
necting string;  and  after  descending  through  a  given 
space  the  4  lbs.  weight  is  detached  and  taken  away  without 
interrupting  the  motion  ;  find  through  what  space  the 
remaining  6  lbs.  weight  will  descend. 

Ans.  Through  J  of  the  given  space. 

18.  Two  weighta  are  attached  to  the  extremities  of  a 
string  which  is  hung  over  a  smooth  pulley,  and  the  weights 
are  observed  to  move  through  6.4  feet  in  one  second  ;  the 
motion  is  then  stopped,  and  a  weight  of  5  lbs.  is  added 
t.)  the  smaller  weight,  which  then  descends  through  the 
same  space  i\»  it  ascended  before  in  the  same  time  ;  deter- 
mine  the  original  weights.  Ans.  |  11)8. ;  V  lbs. 

19.  Find  what  weight  must  Iw  added  to  the  Bnmller 
wcigltt  in  Ex.  5,  so  that  tlie  acceleration  of  the  systen.  may 


over  the  edge  of  a 
t)e.  along  the  table; 
jnaion  of  the  string, 
jr  sec. ;  (2)  20  lbs. 

a    platform ;    find 

the  latter    is  de- 

,  and   (2)  if   it  ia 

I  7  lbs.;  (2)  9  lbs. 

hang  over  a  smooth 
igh  which  they  will 
Ans.  9|  ft. 

es  respectively  hang 
find   the  space  de- 
!  scoouds  from  rest 
»  =  26,  V  =  10. 

.  togethei  pull  one 
by  means  of  a  cou- 
;  through  a  given 
taken  away  without 
B[h   wi)at  space  the 

f  the  given  space. 

10  extremities  of  a 
ey,  and  the  weights 
in  one  second  ;  the 
of  5  lbs.  is  added 
icends  through  the 
same  time ;  deter- 
s.  I  11)8. ;  V  lbs. 

I'd   to   the  Bnjaller 
of  the  syateiL  may 


XX' 


nBS. 


810 


have  the  same  nnmerioal  value  aa  before,  but  may  l)e  in 


the  opposite  directisn. 


Ana. 


20.  A  body  \s.  projected  np  a  rough  inclined  plane  with 
the  velocity  which  would  be  acquired  in  faUing  freely 
through  12  foet,  and  just  reaches  the  top  of  the  plane ; 
the  inclination  of  the  plane  to  the  horizon  is  60°,  and  the 
coeflScient  of  friction  is  equal  to  tan  30°;  find  the  height  of 
the  plane.  Ana.  9  feet. 

21.  A  body  is  projected  up  a  rough  inclined  plane  with 
the  velocity  ^g  ;  the  inclination  of  the  plane  to  the  horizon 
is  30°,  and  the  coeflHcient  of  friction  is  equal  to  tan  15° ; 
find  the  distance  along  the  plane  which  the  body  will 
describe.  ^na.  g  (v^S  +  1). 

22.  A  body  is  projected  up  a  rough  inclined  plane  ;  the 
inclination  of  the  plane  to  the  horizon  is  «,  and  the  coef- 
ficient of  friction  ie  tan  e ;  if  m  be  the  time  of  ascending, 
and  n  the  time  of  descending,  show  that 


\«  /  ~  sin  (a  +  e)' 


23.  A  weight  P  is  drawn  up  a  smooth  plane  inclined  at 
an  angle  of  30°  to  the  horizon,  by  means  of  a  weight  Q 
which  descends  vortictjUy,  the  weights  being  connected  by 
a  string  passing  over  a  small  pulley  at  the  top  of  the  plane ; 
if  the  acceleration  be  one^fourth  of  that  of  a  body  fa]lh.j 
freely,  find  the  ratio  of  ^  to  P.  Ant.  Q  =  P. 

24.  Two  weights  P  and  Q  are  connected  by  a  string, 
and  Q  hanging  over  the  top  of  a  smooth  plane  inclined  at 
30°  to  the  horizon,  can  draw  P  up  the  length  of  the  plane 
in  just  half  the  time  that  P  would  take  to  draw  up  Q ; 
show  that  Q  U,  half  m  heavy  again  as  P. 


830 


MXAMPhaa. 


25.  A  particle  moyes  in  a  straigfat  line  undo*  the  action 
of  an  attraction  varying  inversely  as  the  (|)th  power  of 
the  distance;  sho\»  that  the  velocity  acquired  hy  falling 
from  an  infinite  distance  to  a  distance  a  from  the  centre  is 
equal  to  the  velocity  which  would  be  acquired  in  moving 

from  rest  at  a  distance  a  to  a  distance  ^• 


m 


te  nndear  the  action 
the  (|)th  power  of 
acquired  by  falling 
[  from  the  centre  is 
acquired  in  nioTing 


CHAPTER    II. 

CENTRAL    FORCES.* 

180.  Definitlona. — A  central  force  is  one  which  acts 
directly  towards  or  from  a  fixed  point,  and  is  called  an 
attractive  or  a  repulsive  force  according  as  its  action  on 
any  particle  is  attraction  or  repulsion.  The  flxed  point  is 
called  the  Centre.  The  intensity  of  the  force  on  any  par- 
ticle is  some  function  of  its  distance  from  the  centre. 
Since  the  case  of  attraction  is  the  most  important  af>plica- 
tion  of  the  subject,  we  shall  take  that  as  our  standard  case ; 
but  it  will  be  seen  that  a  simple  change  of  sign  will  adapt 
our  general  formnlsa  to  repulsion.  If  the  centre  be  itself 
in  motion,  we  may  treat  it  as  flxed,  in  which  case  the  term 
"actual  motion  "of  any  particle  means  its  motion  "rela- 
tive" to  the  ct  itre,  taken  as  fixed. 

The  line  firom  the  centre  to  the  particle,  is  called  a 
Radius  Vector.  The  path  of  the  particle  under  the  action 
of  an  attraction  or  repulsion  directed  to  the  centre  is 
called  its  Orbit.^  All  the  forces  of  nature  with  which  we 
are  acquainted,  are  central  forces ;  for  this  reason,  and  be- 
cause the  motion  of  bodies  under  the  action  of  central 
forces  is  a  branch  of  the  general  theory  of  Astronomy,  we 
shall  devote  this  chapter  to  the  consideration  of  their 
action. 

181.  A  PuHele  under  the  Aotion  of  a  Central 
Attnotion;  Reqnired  the  Polar  Equation  of  the 
Path. — The  motion  will  clearly  take  place  in  the  plane 
passing  through  the  centre,  and  the  line  along  which  the 

•  TbiM  chapter  conUln*  the  Ant  prinolplei  of  lUtbWDaticai  Attronomy.     It 
nuy,  howaver,  be  omlltad  bj  the  •tudent  of  KngtMering. 
t  OkU«d  Omtnl  Urbiu. 


:il 


M 


CKlfTSAL  ATTRACnO'cf. 


particle  is  initially  projected,  as  there  is  nothing  to  with- 
draw the  particle  from  it.  Let  the  centre  of  attraction,  0, 
be  the  origin,  and  OX,  OY,  any 
two  lines  iirough  0  ftt  right  angles 
to  each  other,  be  the  axes  of  co- 
ordinates. Let  (x,  y)  be  the 
position  of  the  particle  M  at  the 
time  t,  and  (r,  9)  its  position 
referred  to  polar  co-ordinatos, 
OX  being  the  initial  line.  Then, 
calling  P  the  central  attractive 
force,  we  have  for  the  components  parallel  to  the  axes  of  x 

and  y,*  respectively,  --  P-,  —  P^,  the  forces  being  nega- 
tive, since  they  tend  to  diminish  the  co-ordinates.  There- 
fore the  equations  of  motion  are 


(1) 


Multiplying  the  former  by  y,  and  the  latter  by  x,  and 
iubtracting,  wo  have 

;  — 2  _  «  — 


a;^-y,^  =0. 


(3) 


Integrating  we  have 


.% 


dt       "dt 
where  A  is  .a  undetermined  constant 
Since  x  =5  r  oos  0,  and  y  =  r  sin  0,  we  hav« 
dx  =  COS  B  dr  —  r  Bin  0  do, 
dy  =z  nine dr  +  rcoaO d9, 
which  in  (3)  gives 


(8) 


(*) 


o:y. 


is  nothing  to  with- 
itie  of  attraction,  0, 


Fli.M 

llel  to  tlie  axes  of  x 
e  forooB  being  nega< 
D-ordinates.    There- 

■  Pf.  (1) 

he  latter  by  x,  and 

(3) 


(8) 


ire  iuiye 

ide, 

)d$, 


<*) 


CMlfTBAL  ATTSACnOJf, 

d(  ^  '*• 


898 

(5) 


Again,  multiplying  the  first  and  second  of  (1)  by  2tbe 
and  2dy  respectively,  and  adding,  we  get 


2dxdh:  +  %dy  dhf  _       2P{xdx  +  ydy) 
d0^  ~  r  * 


d 


da* 


,5?  "•■<«« 


-2/Wn 


m 


Substituting  in  (6)  the  values  of  dafi  and  Jy*  from  (4),  we 
have 


(7) 


Put  r  ss  - ;  and  .  •.  dr  as  —  — | ;  and  (7)  becomes 


2P 


./dvF  .A        2P  , 


performing  the  differentiation  of  the  fint  member>  and 
dividing  by  idu,  and  transposing,  we  get 


dht 


+  «♦- 


]?u* 


m 


which  i$  th*  differential  equation  of  the  orbit  described ; 
and  as,  in  any  particular  instance,  the  force  P  will  be  given 
in  terms  of  r,  and  therefore  in  terms  of  u,  the  integral  of 
this  equation  will  be  the  polar  equation  of  the  requijr^ 
path. 
6<dving  (8)  for  P  w«  have 


324 


CENTRAL  ATTRACTtON. 


^  =  »^(^"  +  «)' 


(») 


which  is  the  same  resolt  that  was  found  by  a  different  pro- 
cess in  Art  163  for  the  acceleration  along  the  radius 
vector. 

OoB.  1. — The  general  integ.als  of  (1)  will  contain  four 
arbitrary  constants.  Ona,  A,  that  was  introduced  in  (6), 
and  two  more  will  be  introduced  by  the  integration  of  (8). 
If  the  value  of  r  ia  terms  of  0,  deduced  from  the  integral 
of  (8),  be  substituted  in  (5),  and  that  equation  be  then 
integrated,  the  fourth  constant  will  be  introduced,  and  the 
path  of  the  particle  and  its  position  at  any  time  will  be 
obtained.  The  four  constants  must  be  determined  from 
the  initial  circumstances  of  motion;  viz.,  the  initial 
position  of  the  particle,  depending  on  two  independent 
co-ordinates,  its  initial  velocity,  and  its  direction  of  pro- 
jection. 

Cor.  a.— By  means  of  (9)  we  may  ascertain  the  law  of 
the  fcAse  which  must  act  upon  a  particle  to  cause  it  to 
describe  a  given  curve.  To  eflfect  this  we  must  determine 
the  relation  between  v,  and  B  from  the  polar  equation  of  the 
orbit  referred  ^  >  the  required  centre  as  pole  ;  we  must  then 
differentiate  .  twice  with  respect  to  0,  uid  substitute  the 
result  in  the  expression  for  P,  eliminating  9,  if  it  occurs, 
by  means  of  the  relation  between  «  and  9.  In  this  way  we 
shall  obtain  P  in  terms  of  u  alone,  and  therefore  of  r 
alone. 

OoB.  8.— When  we  know  the  relation  between  r  and  fl 
from  (9),  we  may  by  (6)  determine  the  time  of  describing 
a  given  portion  of  the  orbit ;  or,  conversely,  find  the  posi- 
tion of  the  particle  in  its  orbit  at  any  time.* 

*  IM  lUt  ud  StMle'i  DyuuniM  of  >  I^wtkla,  p.  IM;  alw  Pm(t'«  Ifack't, 


V. 

;  (9) 

by  a  different  pro- 
along  the  radius 

I  will  contain  four 
introduced  in  (5), 
integration  of  (8). 
from  the  integral 
equation  be  then 
itrodnced,  and  the 
;  any  time  will  be 
I  determined  from 
viz.,  the  initial 
two  independent 
I  direction  of  pro- 
certain  the  law  of 
;icle  to  canse  it  to 
re  must  determine 
liar  equation  of  the 
ole  ;  we  must  then 
and  aubstitnte  the 
ng  6,  if  it  occurs, 
>.  In  this  way  we 
ind  therefore  of  r 


I  between  r  and  6 
time  of  describing 
sly,  find  the  posi- 
ic.* 

lU;  atao  Pmtt'*  Ifach't, 


THB  BBCTIONAL  AREA. 


OoB.  4, — If  p  is  the  perpendicular  from  the  origin  to 
the  tangent  we  have  from  Oalcnlus,  p.  176, 


xdy  —  ydx  =:  pd9  i 


which  in  (3)  gives 
and  this  in  (6)  gives 


ds 
dt 


P' 


(10) 


,h* 


Differentiating,  and  solving  for  P,  we  have 

A»rfr 
f  dp' 


(11) 


which  is  the  equation  of  the  orUt  between  the  radius  vector 
and  the  perpendicular  on  the  tangent  at  any  point. 

182.  The  Etoctorial  Area  Swept  over  by  tiie 
Raditis  Vector  of  the  Particle  In  any  time  is  Pro- 
portional to  the  Time. — Let  A  denote  this  area ;  then  we 
have  £rom  Calculus,  p.  364, 


if  A  and  t  be  both  measured  from  the  commencement  of 
the  motion.  Therefore  the  areas  swept  over  by  the  radius 
vector  in  different  times  are  proportional  to  the  times,  and 
equal  areas  will  be  described  in  equal  times. 

Cob. — If  t  =  1,  we  have  A  =  |A.    Hence  h  =  twice 
the  sectorial  arja  described  in  one  unit  of  time. 

1B3.  The  Velocity  of  the  Particle  at  any  Point 
of  ita  OrMt— We  have  for  the  velooity, 


A  =  i/r»rfd 

=  ^fhdt,hj  (5)  of  Art.  181, 

i- 

=  iAf, 

1 

886       vsLocmr  at  ant  poi^rr  or  rmi  obbit. 


=  j^  by  (10)  of  Art  181.       (1) 

Hence,  the  velocity  of  the  patticle  at  each  point  of  its 
path  is  inversely  proportional  to  the  perpendicular  from  the 
centre  ott  the  tangent  at  that  point. 

OoB.  1.— We  have,  by  Calculus,  p.  180, 

2-  _  1       1   rff* 

=  «»  +  ^',  since  f  =  -  (Art.  181), 


which  in  (1)  gives 

•'=1  =  »•(«•+ a. 

another  important  expression  for  the  velocity. 
Cob.  2.— From  (6)  of  Art.  181,  we  have 


(2) 


(8) 


I*t  V  be  the  velocity  at  the  point  of  projection,  at 
which  let  r  =:  B,  and  since  P  is  some  function  of  r,  let 
P  =  /(r),  then  integrating  (3)  we  get 

which  is  another  expression  for  the  velocity ;  and  since  this 
is  a  ftinotion  only  of  the  corresponding  distance*,  R  and  r, 
it  follows  that  th$  velocity  at  any  point  qf  tho  orbit  is 


rjfV  OBBJT. 


of  Art  181. 


(1) 


t  each  point  of  its 
pendicular  from  the 


30, 


36  f  =  -  (Art  181), 


). 


(2) 


locity. 

IT© 

Fdr. 


(8) 


t  of  projection,  at 
e  fanotion  of  r,  let 


(r)], 


{*) 


sity ;  and  since  this 
diatanoM,  E  and  r, 
nnt  qf  the  orbU  %$ 


vBLOcrtr  AT  Atrr  potsr  of  wr«  oMBtt.      88? 

independent  of  the  path  described,  and  dtpmdt  iokljf  on  the 
magnitude  of  the  atttaetion,  the  dietanoe  of  th$  point  from 
the  centre,  and  the  velocitif  and  distance  of  projection. 

From  (4)  it  appears  that  the  velocity  is  the  same  at  all 
points  of  the  same  orbit  which  are  equally  distant  from  the 
centre ;  if  r  =:  i?,  the  velocity  =  V;  and  thus  if  the  orbit 
is  a  reentering  cunrc,  the  particle  always,  in  its  successive 
revolutions,  passes  through  the  same  point  with  the  same 
velocity. 

If  the  velocity  vanishes  at  a  distance  a  from  the  centre 
(4)  becomes 

t;»  =  3[A(a)-A(r)]  (6) 

and  a  is  called  the  radius  of  the  circle  of  zero  Velocity. 


OoB.  3.- 

-From  (3)  we  have 

rf(r»)  =  - 

2Pdr; 

.*.    vdv  =  — 

Pdr. 

Taking  the 

logarithm  of  (1)  we 

have 

log  w  =  log  A 

-log/. 

Differentiating  we  get 

dv 

P 

(«) 


(*J 


Dividing  (6)  by  (7),  we  get 


=  2P  X  J  chord  of  curvature*  through  the  centre ;  (8) 


•  To  prove  that  f  ~  U  one-f onrth  the  cbord  of  OWmttM. 

8  dp 
Let  MD  (Fig.  81),  ba  tbe  Ungent  to  the  orbit,  end  C  the  MDtre  of  cnrratara ;  let 
OD  =|),  CM  =  p,  tbe  radliM  of  curvature ;  end  the  angle  HSS  -  ^.   Then  MS,  the 


338 


VSLOCJTT  AT  Afrr  POINT  OP  THE  OBBIT. 


and,  comparing  this  with  (6)  of  Art.  140,  it  appears  that 
the  particle  at  finy  point  has  the  same  velocity  which  it 
would  have  if  it  mov^  from  rest  at  that  point  towards  the 
centre  of  force,  under  the  action  of  the  force  continuing 
constant,  through  one-fourth  of  the  chord  of  the  circle  of 
curvature. 

Hence,  the  velocity  of  a  particle  at  any  point  of  a  central 
orbit  is  the  same  as  that  which  would  be  acquired  by  a 
particle  moving  freely  from  rest  through  one-fourth  of  the 
ciiord  of  curvature  at  that  point,  through  the  centre,  under 
the  action  of  a  constant  force  whose  magnitude  is  equal  to 
ihat  of  the  central  attraction  at  the  point. 

Cob.  4. — If  the  orbit  is  a  circle  having  the  centre  of  force 

part  of  the  radlns  vector  CM,  which  la  intercepted  by  the  circle  of  cnrratON  is 
Called  the  chord  tf  curvature.    Ita  valae  is  determined  as  follows ; 


We  have  (Fig.  81) 


*  =  fl  +  OMD 
=  »  +  sln-i 


rVr'—tf 
Viom  Calcoluf,  p.  180,  (10),  we  have 

pdr 


a$  =  — 


r  Vr"  — p" 


and 

Subatitiitlng  CI)  in  (1)  we  get 


<$,  =  '■ 


rOr 


<l* 


P 

_       dp 


yfr'  — p* 


But  Oalcalns,  p.  381,  we  bare 


dr 


Now  MS  (Tig.  81)  =  8M0  sin  OMD, 


=  ^?  =  *|.hy(5) 


=  the  chord  of  cnrratiue ;  therefore 

tap 


?  :j-  =  one-fourth  the  chord  of  carvatniv. 


« 


m 


(«> 


m 


HE  OBBIT. 

10,  it  appears  that 

velocity  which  it 

point  towards  the 

I  force  coutiDuing 

rd  of  the  circle  of 


point  of  a  central 
'  be  acquired  by  a 
i  one-fourth  of  the 
h  the  centre,  under 
titude  is  equal  to 

the  centre  of  force 

tbe  circle  of  cnmtara  ia 
FoUowb; 


0) 
W 


tare. 


TBB  ORBIT  mrDER    VARtABLB  ATTRACTION. 


320 


in  the  centre,  and  R,  V,  P,  are  reapeotiyely  the  radius, 
velocity  and  central  force,  we  have 

r»  =  PR. 


OOR.  5.— From  (5)  of  Art  181,  we  have 


de 

dt 


(9) 


The  first  number,  being  the  actual  velocity  of  a  point 
on  the  radius  vector  at  the  unit's  distance  from  the  centre, 
is  the  angular  velocity  of  the  particle  (Art.  160).  Hence 
the  angular  velocity  of  a  particle  varies  inversely  as  the 
square  of  the  radius  vector. 

ScH.— A  point  in  a  central  orbit  at  which  the  radius 
vector  is  a  maximum  or  minimum  is  called  an  Apse ;  the 
radius  vr^tor  at  an  apse  is  called  an  Apsidal Distance ;  and 
the  angle  between  two  consecutive  apsidal  distances  is  called 
an  Apsidal  Angle  of  the  orbit    The  analytical  conditions 

for  an  apse  are,  of  course,  that  ^  =  0,  ai.d  that  the  first 

derivatiye  which  does  not  vanish  should  be  of  ati  even 
order.  The  first  condition  ensures  that  the  radius  vector 
at  an  apse  is  perpendicular  to  the  tangent. 

184.  The  Orbit  when  fhe  Attraction  Vaxiee  In- 
v-ersely  as  the  Square  of  the  Diatanca— J  particle  is 
projected  from  a  given  point  in  a  given  direction  with  a  given 
velocity,  and  moves  under  the  action  of  a  central  attraction 
varying  inversely  as  the  square  of  the  distance  ;  to  determit.e 
the  orbit. 

Let  the  centre  of  force  be  the  origin;  V  =  the  velocity 
of  projection  ;  i?  =r  the  distance  of  the  point  of  projection 
from  the  origin ;  /3  =  the  angle  between  R  and  the  line  of 


Hi 


880       TBlt  OBBIT  Umxn   VABIABhW  ATTBACTIOB. 

projeotioii ;  and  let  ^  =  the  absolate  force  and    <  s=  0 
when  the  particle  is  projected.    Then  since  the  relooity  s 

-  (Art  183),  and  at  the  point  of  projection  />  =  ^  sin  /3, 

wc  have 

K  Sin  p 

As  the  force  yaries  inversely  as  the  square  of  the  distance, 
wehaye 

p  SB  ^  SB  fiv^,  ^since »"  =  z)-  (2) 


which  in  (9)  of  Aft  181  gives 

Multiplying  by  ddu  and  integrating,  we  get 

S  +  «*  =  2?.«  +  C; 


(3) 


d0» 


¥ 


1       1  du*  F* 

when  «  =:  0,  tt  =±  -  i=  -g,  and  ^  +  «» sa  -p-,  (Art  183, 

Cor.l);  therefore 


'  ~  A»       A»i2 


A»i? 


Substituting  thl«  valtte  for  c  We  get 


Therefore  (Art.  183,  Cor.  1)  we  have 

(velocity)'  =  F»  +  2^  (^  -  ^ 


(4) 


(5) 


'  ATTBACTIOir. 

te  force  aod    <  r=  0 
since  the  Telocity  =z 

jection  />  =  i2  sin  /3, 


f2  sin  /3. 


(1) 


qnare  of  the  distance, 

(8) 

re  get 

.tt»s=-^,  (Art.188, 
-2p 


»i2 


8p« 


-i) 


w 


0) 


TJ»  OSBtT  tniJ>MB  VABtABLM  ATfttAOTtOlf.       381 

which  showa  that  the  velocity  it  th«  grmtest  whm  ritth* 
least,  and  the  leaet  when  r  is  the  greatest. 
Changing  the  fonn  of  (4)  we  Uave 


d0* 


-       h»R       ■^h*-\h*-'V' 


(6) 


To  express  this  in  b  bimpler  form,  let 

g  =  *,  and  ^^^~  +  ^  =  <^.;  a^^  (6)  becomes 
_=c!-.(«-J)a; 


d9» 


—  du 


[c»  -.  (»  -  J)«]* 


=  dO, 


the  negatiTe  sign  of-  the  radical  being  taken.    Integrating 
we  have. 


COS" 

where  c'  is  an  arbitrary  constant; 

.'.     tt  =a  4  +  C  DOS  («  —  C*). 


(7) 


Replacing  in  (7)  the  values  of  h  and  c,  and  the  value  of  h, 
from  (1),  and  dividing  both  terms  of  the  second  number  by 
Hf  we  have  for  the  equation  of  the  path. 


1  +  ri  (  F»iJ  -  8^)  /?  r»  Bin»  /3  +  1 J  008  (»-«') 
ti  = 1 _____^___- , 


^r»8in»/J 


(8) 


which  is  the  equation  of  a  conio  section,  the  pole  being  at 
the  focus,  and  the  angle  (9  —  c')  being  measared  from  the 


THE  ORBIT  UNDER    VARIABLE  AVtRACTION. 

shorter  length  of  the  axis  major.  For  if  0  is  the  eccentricity 
of  a  conic  section,  r  the  focal  radias  vector,  and  ^  the 
angle  between  r  and  that  point  of  a  conic  section  which  is 
nearest  the  focus,  we  hare, ' 


1  1  +  c  cos  A 

—  H*        ta^m        '     ■         ■  ■  '■       -  • 


Comparing  (8)  aud  (9),  we  see  that 


fl»  =  -(FJ72-2^)i?r«3in»/3  +  1; 


4»  =  d 


e\ 


(») 

(10) 
(11) 


Now  the  couis  section  is  an  ellipse,  parabola,  or  hyper- 
bola, according  as  e  is  less  than,  equal  to,  or  greater  than 
unity ;  and  from  (10)  e  is  leas  than,  equal  to,  or  greater 
than,  unity  according  as  V*B  —  2fi  is  negative,  zero,  or 
positive ;  therefore  we  see  that  if 


2n 


F*  <  -^,  e  <  1,  and  the  orbit  is  an  ellipse, 


2u 
F»  =  -p,  e  =  1,  and  the  orbit  is  a  parabola. 


(12) 
(13) 


2u 
F»  >  ~,  «  >  1,  and  the  orbit  is  a  hyperbola.     (14) 

CoR.  1.— By  (1)  of  ;^rt.  173,  we  see  that  the  square  of 

the  velocity  of  a  particle  falling  from  infinity  to  a  distance 

Ji  from  Ihe  centre  of  force,  for  the  la^  of  atti-action  we 

, ,    .       .    2« 
arc  considering,  is  -^.    Hence  the  above  conditions  may 

be  expressed    more  concisely   by  saying  that   the  orbit, 
d^mihed  about  this  centre  0/  force,  wiil  be  an  eUipse,  a 


Mfe 


tirRACfioir. 

6  is  the  eccentricity 
vector,  and  ^  the 
aic  section  which  is 


(») 


*/3  +  l; 


(10) 
(11) 


parabola,  or  hyper- 

to,  or  greater  than 

iqnal  to,  or  greater 

1  uogative,  zero,  or 


an  ellipse,         (12) 


a  parabola,       (13) 


a  hyperbola.     (14) 

that  the  square  of 
finity  to  a  distance 
h^  of  atti'action  we 

ve  conditions  may 

ng  that    the  orbit, 
ill  be  an  ellipse,  a 


TBS  ORBIT  AN  BLLIPSJB. 


883 


parabola,  or  a  hyperbola,  according  as  the  velocity  is  ksa 
than,  equal  to,  or  greater  than,  the  velocity  from  infinity. 

The  species  of  conic  section,  therefore,  does  not  depend 
on  the  position  of  the  line  in  which  the  particle  is  pro- 
jected, but  on  the  velocity  of  projection  in  reference  to  the 
distance  of  the  point  of  projection  fh)m  the  centre  of 
force. 

Oor.  2.— From  (11),  we  see  that  6  —  c'  is  the  angle 
between  the  focal  radius  vector, 
r,  and  that  part  of  the  principal 
axis  which  is  between  the  focus 
and  the  point  of  the  orbit  which 
is  nearest  to  the  focus ;  i.  e.,  it 
is  the  augle  PFA  (Fig.  82) ;  and 
therefore  if  the  principal  a^Js  is  the  initial  line  c'  =  0. 

185.  Suppose  the  Orbit  to  be  an  EUipse.— Here 

p-s  ^  ^ .  eo  that  from  (10)  we  have 

c»  =  1  -  i  (2?*  -  F»/Z)  i?  r»  sin» /3. 


(1) 


Now  the  equation  of  an  ellipse,  where  r  is  the  focal 
radius  vector,  9  the  angle  Iwtween  r  and  the  shorter  seg- 
ment of  the  miyor  axis,  2o  the  major  axis,  e  the  eccon- 

'■""  1  -f-  eome* 


1  fl  Cos  0 

•■*    »*  =  o  (1  _  «»)  +  ^JT^  e») ' 

comparing  (2)  with  (8)  of  Art,  184,  we  have 

1 ± . 


(») 


4 


834 


TBB  ORBIT  Air  SLUPSK. 


substituting  f or  1  —  «»  its  yaliw  from  (1),  and  aolTingfor 
a,  we  have 

"  -  2^  _  yJR' 


(8) 


which  shows  that  ths  major  axis  is  independent  of  the  direc- 
tion of  projection. 

We  may  explain  the  several  quantities  which  we  have 
used,  by  Fig.  82. 

B  is  the  point  of  projection;  FB  =  5;  DB  is  the  line 
along  which  the  particle  is  projected  with  the  velocity  V; 
FBD  =  p,  the  angle  of  projection;  FP  =  r;  PFA  =  d; 
FD  =  /2  «n  /3;  if  (9  =  90°,  the  particle  is  projected  from 
an  apse,  i.  «.,  from  A  or  A'. 

Cob.  1.~To  determine  the  apsidal  diitances,  FA  and 
FA',  we  must  put  ^  =  0,  (Art  183,  8oh.),  and  (4)  of 
Art  184  give  us  the  quadratic  equation 


(*> 


the  two  roots  of  which  are  the  reciprocals  qf  the  two  apsidal 
diitances,  a{l  —e)  and  a  (1  -f  e). 

Cor.  a.— Since  the  coefficient  of  the  second  term  of  (  ) 
is  the  sum  of  the  roots  with  their  signs  changed,  we  have 


a(l-e)  ^  a^l-e) 


«(l-^=^'l 


which  gives  the  latus  rtetum  of  the  «rUi. 


(») 


rj& 


(1),  and  lolTing  for 


(3) 
mdeni  of  the  direc- 


es  whioh  we  have 

2;  DB  is  the  line 
ith  the  velocity  V; 

=  r;  PFA  =  6; 

is  projected  from 


ittanccB,  FA  and 
Soh.),  and  (4)  of 

:  0.  (4) 

jff  the  two  apaidal 


econd  term  of  \  > 
langed,  we  have 

in. 


(») 


MBPLgR'S  LAWS. 

CJOB.  3.— From  Art  183  we  have,  calling  Tthe  time, 


T=: 


aA 


where  A  is  the  area  swept  over  by  the  radius  vector  in  the 
time  T.    Therefore  for  the  time  of  describing  an  ellipse, 

we  have 

_      a  area  of  ellipse 


2na*  Vl  —  e« 


,  from  (5), 


Sff 


Vf' 


which  is  the  time  oceupiad  bif  the  particle  in  passing  from 
any  point  of  the  ellipse  around  to  the  same  point  cgain.* 

l^'i.  KflplftX**  Xiaws. — By  laborious  calculation  from 
an  immense  series  of  observations  of  the  planets,  and  of 
Mars  in  ptsrticular,  Kepler  enanoiattd  the  following  as  the 
Ifiws  of  the  planetary  motions  aboat  the  Sun. 

/,  'A%e  orbits  of  ihe  planed  are  ellipses,  of  whioh 
the  Sun  occupies  a  focus. 

II.  The  radius  vector  of  each  planet  describes 
equal  areas  in  equal  times. 

III.  The  squares  of  the  periodic  times  of  the 
planets  are  a<i  Ine  cubes  of  the  ,naJor  axes  of  their 
orbits. 

187.  To  Detannin«  the  Mature  of  the  force  which 
Acta   apon  the   Planetary  Byatem.— (1)  ^^"^  ^''" 

•  oidlad  AHmHo  nmt. 


^^ 


336 


PLAJfBTASr  srsTxir. 


second  of  these  laws  it  follows  that  the  planets  arc  retained 
in  their  orbits  hy  fm  attraction  tending  to  the  Snn. 

Let  {x,  y)  he  the  position  of  a  planet  at  the  time  t 
referred  to  two  co-ordinate  axes  drawn  through  the  Sun  in 
the  plane  of  motion  of  the  planet ;  X,  Y,  the  component 
accelerations  due  to  the  attraction  acting  on  it,  resolved 
parallel  to  the  axes ;  then  the  equations  of  motions  are 


^  _ 


Y; 


yX 


(1) 


Bnt,  by  Kepler's  second  law,  if  A  be  the  area  described 
by  the  radius  vector,  -^  is  constant, 


rfA           ^r*d6 
"    dt    ''    ^  di 

^♦(''f -yfl^*^"***"*- 

DifliBrentiating,  we  have 

d»y         dh!  __ 
''dfi       ^dfl-^' 

.'.    xY  —  ifX  =  0,  from 

(1). 

which  shows  that  the  axial  components  of  the  acceleration, 
due  to  the  attraction  acting  on  the  planet,  are  proportional 
to  the  co-ordinates  of  the  planet;  and  therefore,  by  the 
parallelogram  of  forces  (Art.  30),  the  resultant  of  X  and  Y 
puMes  through  the  origin. 


W^W  V*-^  *K»5SB?*; 


planets  arc  retained 
to  the  Snn. 
[anet  at  the  time  t 
hrough  the  Sun  in 
V,  the  component 
ing  on  it,  resolved 
of  motions  are 


yx  (1) 

the  area  described 


Qstant 


a(l), 


3f  the  acceleration, 
it,  are  proportional 
I  therefore,  by  the 
lultant  of  Jr  and  F 


PLANBTART  araTJBM. 


887 


Hence  the  forces  acting  on  the  planets  aU  pass  throiigh 
the  Sun's  centre. 

(2)  From  the  first  of  these  laws  it  follows  that  the 
central  attraction  varies  inversely  as  the  square  of  the 
distance. 

The  polar  equation  of  an  ellipse,  referred  to  its  focus,  is 

__  o(l-e') 

**  "^  1  +•  «  cos  ©' 


or 


Hence 


u  = 


+ 

1  +  e  cos  0 
a(%-^' 

1 


^  , 

rfe»  ■*■  **  ~  a  (1  -  fl») ' 


and  therefore,  if  P  is  the  attraction  to  the  focus,  we  have 
[Art  181,  (9)], 


h» 


~  a  (1  -  e»)  r» 

Hence,  if  the  orbit  he  an  ellipse,  described  about  a  centre 
of  attraction  at  the  focus,  the  law  of  intensity  is  that  of  the 
inverse  square  of  the  distance. 

(3)  From  the  third  law  it  follows  that  the  attraction  of 
the  Sun  (supposed  fixed)  which  acts  on  a  unit  of  mass  of 
each  of  the  planets,  is  the  same  for  each  planet  at  the  same 
distance. 

By  Art.  186,  Cor.  3,  we  have 


in* 


15 


S?8 


KXAMPLMB. 


But  by  the  third  kWj  T»  «  o«,  and  therefore  )«  tottst  be 
constant ;  i.  e.,  the  strength  of  attraction  of  the  Sun  must 
be  the  same  for  all  thi  planets.  Hence,  not  only  is  the  law 
of  force  the  same  for  all  the  planets,  but  the  abaotuU  force 
is  the  same. 
This  very  brief  discussion  of  central  forces  is  all  that  we 
II'  have  space  for.    To  pursue  these  enquiries  farther  would 

compel  us  to  omit  matters  that  are  more  especially  entitled 
to  a  place  in  this  book.  The  student  who  wishes  to  pursue 
the  study  further  is  referred  to  Tait  and  Steele's  Dynamics 
of  a  Particle,  or  Price's  Anal.  Mech's,  Vol  I,  or  to  any 
work  on  Mathematical  Astronomy.  We  shall  concludD 
with  the  following  examples. 


EXAMPLES. 

1.  A  particle  describes  an  ellipse  under  an  ftttraction 
always  directed  to  the  centre ;  it  is  required  to  find  the  law 
of  the  attraction,  the  velocity  at  any  point  of  the  orbit,  and 
the  periodic  time. 

(1)  The  polar  equation  of  the  ellipse,  the  pole  at  the 
centre,  is 

cos»0      sin'O 
a*    "•■     i»    ' 


«f> 


(1) 
•  •    ""dd  =  iji  -  aJ  sine  cos  »,  (») 

But  [Art.  181,  (9)]  we  have 


MXAMPLWa, 


lerefore  ju  tiiiist  be 

a  of  the  Snn  must 

not  only  is  the  law 

t  the  abaotule  force 

)rce8  is  all  that  we 
iries  farther  would 
3  especially  entitled 
bo  wishes  to  pursue 
Steele's  Dynamics 
,  VoL  I,  or  to  any 
We  shall  conclude 


nder  an  attraction 
ired  to  find  the  law 
at  of  the  orbit,  and 

• 

e,  the  pole  at  the 

(1) 
J  9,  (2) 

-  Bin'  6).         (3) 


by  (3), 

(COS*  e  —  sin*  d)],  by  (2), 

by  factoring, 


A*    1     1     u    /iv  **  - 


m 


and  therefore  the  attraction  varies  directly  as  the  distance. 
If  ^  =  the  absolute  force  we  have,  by  (4), 

A»  =  ^a»ft».  (6) 

(2)  If  t;  =  the  velocity,  we  have,  by  Art.  188, 

t»»  =  ^  =  ^J  (Anal.  Geom.,  p.  133) 

=  tib\  by  (6), 
where  V  is  the  semi^diameter  conjugate  to  t. 
.'.    V  tsh'  V/ii. 

(3)  If  2*  =  the  periodic  time,  we  have,  by  Art.  182, 

_      2iTfl&        %n 


V^ 


,  by  (6), 


and  hence  the  periodic  time  is  independent  of  the  magni- 
tude of  the  ellipse,  and  depends  only  on  the  absolute 
central  attraction.    (;See  Tait  and  Steele's  Dynamics  of  a 


840 


BXAMPLSa. 


Particle,    p.    144,    also    Prioe'fl   AnaL   Mech's,    Vol.    I, 
p.  516.) 

%.  A  particle  describes  an  ellipse  ander  an  attraction 
always  directed  to  one  of  the  foci ;  it  is  required  to  find  the 
law  of  attraction,  the  velocity,  and  the  periodic  time. 


(1)  Here  we  have 

1  +  6  cos  0 
0(1 -c)»  ' 


u  = 


and 


tPu 
dp 


du  _ 
dd  -  ' 

ecos  9 


e  sin  0 


(1) 


a(l-e»)' 
which  in  (9)  of  Art.  181  gives 


a(l  -fl»)  ~  o(l-c»)    r»' 


(2) 


hence  the  attraction  varies  inversely  as  the  square  of  the 
distance.    Ufi  =  the  absolute  force,  we  have  by  (2) 


(2)  By  Art  183,  Cor.  1,  we  have 

1  -  ,  dM»        20W-1     .     ... 

j^  =  *^'  +  ^  =  5Mrr^'*'y<^>' 

.       h*       ^(2o«  — 1)    .     ,_.       ,  ... 
•  *•    *^  =  p  =      ~~S '    y  ^^^  ^^^  ^*^' 

(3)  If  r  =  the  periodic  time  we  have  (Art.  182) 

y^2rroMl-<^)* 


-  g^g*  (1  --  <**)*  _  ^  i 
"[^(l-e»)]*~  v^"' 


(3) 

(4) 
(6) 


(6) 


.   Mech's,    Vol.    I, 

ander  an  attraction 
required  to  find  the 
periodic  time. 


e  Bin  0 


(1) 


_    i 


(8) 


,8  the  square  of  the 
e  have  by  (2) 

(3) 


ry^JWi  (4) 

3)  and  (4).  (6) 

e  (Art.  182) 


3ff    1 
=  — a*, 


(8) 


SXAMPLSA 


841 


and  hence  the  periodic  time  varies  aa  the  square  root  of 
the  cube  of  the  major  axis. 

3.  Find  the  attraction  by  which  a  particle  may  describe 
a  circle,  and  also  the  velocity,  and  the  periodic  time,  (1) 
when  the  centre  of  attraction  is  in  the  centre  of  the  circle, 
and  (2)  when  the  centre  of  attraction  is  in  the  circum- 
ference. 

(1)  Let  a  =  the  radiiu;  then  the  polar  equation,  the 
pole  at  the  centre,  is 


Also 


1     rfu      <i»M       _ 
r  =  a;    .-.    u  = -;  ^  =  ^  ^- 0; 

.       A2  ,     _      27ra» 

v»  =  -i,    and     T=  -j-» 


(1) 
(8) 


From  (1)  and  (2)  we  have 


"=1' 

and  hence  the  central  attraction  is  equal  to  the  square  of 
the  velocity  divided  by  the  radius  of  the  circle.* 


and 


(2)  The  equation,  is 

r  =  2a  cos  6 ; 


t*  + 


P  :=  8a»A  V  = 


.  • .    2au  =  sec  d, 
=  8a V; 
8a»A« 


and  hence  the  attraction  varies  inversely   as   the   fifth 

•  Called  tbe  OUUrVugal  9bre$.    Sm  Art.  IflS. 


343 


BXAMPLSa. 


powOT  of  the  distance  ;  and  if  ju  =  the  absolute  force,  we 
have  fi  =  8aW; 


h>  = 


and    «•  = 


If  T 


the  periodic  time,  we  have 


fir* 


2« 


na' 


(See  Priced  Anal.  Mech.,  Vol.  Ill,  p.  518.) 


4.  Find  the  attraction  by  which  a  particle  may  deseribe 
the  lemnisoate  of  Bernouilli  and  also  the  velocity,  and  the 
time  of  describing  one  looj^i,  the  centre  of  attraction  being 
in  the  centre  of  the  lemn ideate,  and  the  equation  being 
r»  =  c»  cos  3ff. 


Ans.  P  = 


«2 —'    T 


5.  Find  tlie  attraction  by  which  a  particle  may  describe 
the  cardioid  and  also  the  velocity,  and  the  periodic  time, 
the  equation  being  r  =  a  (1  +  cos  0). 


(>.  Find  the  attraction  by  which  a  particle  may  describe 
a  i)arabola,  and  also  the  velocity,  the  centre  of  attraction 

being  at  the  focus,  and  the  equation  being  r  =  = i« 


Compare  (13)  of  Art.  184. 


7.  Find  the  attraction  by  which  a  particle  may  describe 
a  hyperbola,  and  the  velocity,  the  centre  of  attraction  being 

at  the  focus,  and  the  equation  being  r  =  z — ^• 

'  **  1  +  e  cos  © 


Ans.  P  = 


_  JL_  \.  ^-  M(2a«  +  1) 
a  ( 1  -  e»)  fS '  *^  ~  a 


solute  force,  we 


III,  p.  518.) 

le  may  describe 
jlocity,  aud  the 
.ttraction  being 
equation  being 

e  may  describe 
periodic  time, 

=  (Sfia')K. 

le  may  describe 
9  of  attraction 

*"  ~"  1  +  cos  e' 
)  of  Art.  184. 

e  may  describe 
ittraction  being 

(g*-!) 
■f  fl  cos  6 

o 


MAAMPl^MS. 


343 


8.  K  the  centre  of  attraction  is  at  the  centre  of  the 
hyperbola,  find  the  attraction,  and  velocity,  the  equation 

.    .       eos^  9      sin*  d 
Deing  — ^  ^ 


=  «». 


AtU.    P  =z 55*'  = 


■fir',  v»  =r  ^  (f» ->  a»  +  ¥). 


9.  Find  the  attraction  to  the  pole  under  which  a  particle 
will  describe  (1)  the  curve  whose  equation  is  r  =  2a  cos  nd, 

and  (2)  the  curve  whose  equation  is  r  = -"-     - -. 

1  —  e  cos  hS 


(1  -  n»)  A» 


n»)A». 


That  is,  the  attraction  in  the  first  curve  varies 

partly  as  the  inverse  fifth  power,  and  partly  as  the  inverse 
cube,  of  the  distance ;  and  in  the  second  it  varies  partly  as 
the  inverse  square,  aud  partly  as  the  inverse  cube,  of  the 
distance. 

10.  A  planet  revolved  round  the  sun  in  an  orbit  with  a 
major  axis  four  times  that  of  the  earth's  orbit ;  determine 
the  periodic  time  of  the  planet.  Ans.  8  years. 

11.  If  a  satellite  revolved  round  the  earth  close  to  its 
surface,  determioe  the  periodic  time  of  the  satellite. 

Ana.  ——J  of  the  moon's  period. 

13.  A  body  deecrihefl  an 'ellipse  under  the  action  of  a 
force  in  a  focus  :  compare  the  velocity  when  it  is  nearest 
the  focus  with  its  velocity  when  it  is  furthest  from  the 
focus. 

Am.  As  1  +  c  :  1  —  c,  where  e  is  the  eccentricity. 

13.  A  body  describes  an  ellipse  under  the  action  of  a 
force  to  the  focus  8',  if  if  be  the  other  focus  show  that  the 


844 


MXAMPLES. 


velocity  at  any  point  P  may  be  resolved  into  two  velocities, 
respectively  at  right  angles  to  8P  and  HP,  and  each  vary- 
ing as  HP. 

14.  A  body  describes  an  ellipse  under  the  action  of  a 
force  in  the  centre :  if  the  greatest  velocity  is  three  times 
the  least,  find  the  eccentricity  of  the  ellipse.  Ans.  |  \/2. 

15.  A  body  describes  an  ellipse  under  the  action  of  a 
force  in  the  centre :  if  the  major  axis  is  20  feet  and  the 
greatest  velocity  20  feet  per  second,  find  the  periodic  time. 

Ans.  It  seconds. 


16.  Find  the  attraction  to  the  polo  under  which  a  par- 
ticle may  describe  an  equiangular  spiral.       ^^^^^ 


17.  M  P  =  ^  (5r»  +  8c»),  and  a  particle  be  projected 
from  an  apse  at  a  distance  c  with  the  velocity  firom  infinity ; 
prove  that  the  equation  of  the  orbit  ia 

r  =  |(c«'-e-«'). 

18.  M  P  =  2fi  iX  —  ^,  and  the  particle  be  projected 
from  an  apse  at  a  distance  a  with  velocity  — ,  prove  that 


it  will  be  at  a  distance  r  after  a  time 


+  y/i»  —  a* 


-f  r 


Vh"^^)' 


!s«)SW!?wse?»«s»wiSMP«w»«*«^ 


i  into  two  velocities, 
HP,  and  each  vary- 

tder  the  action  of  a 
locity  is  tliree  times 
llipse.  Ana.  |  ■>/2. 

nder  the  action  of  a 

i  is  20  feet  and  the 

id  the  periodic  time. 

Ans.  It  seconds. 

under  which  a  par- 

al.        .        »     1 
Ana.  P<^^' 

particle  be  projected 
relooity  firom  infinity ; 


particle  be  projected 
ocity  —,  prove  that 


r  Vr»  —  en' 


CHAPTER     III. 

CONSTRAINED    MOTION, 

188.  Definitioiis. — A  particle  is  constrained  in  its  mo- 
tion when  it  is  compelled  to  move  along  a  given  fixed  curve 
or  surface.  Thus  far  the  subjects  of  motion  have  been 
particles  not  constrained  by  any  geometric  conditions,  but 
free  to  move  in  such  paths  as  are  due  to  the  action  of  tho 
impressed  forces.  We  come  now  to  the  ctvse  of  tho  motion 
of  a  particle  which  is  constrained ;  that  is,  in  which  the 
motion  is  subject,  not  only  to  given  forces,  but  to  unrieter- 
mined  reactions.  Such  cases  occur  when  the  particle  is  in 
ft  small  tube,  either  smooth  or  rough,  the  bore  of  which  is 
supposed  to  be  of  the  same  size  as  the  particle ;  or  when  a 
small  ring  slides  on  a  curved  wire,  with  or  without  friction ; 
or  when  a  particle  is  fastened  to  a  string,  or  moves  on  a 
given  surface.  If  we  substitute  for  tho  curve  or  surface  a 
force  whoso  intensitj'^  and  direction  are  exactly  equal  to 
those  of  the  reaction  of  the  curve,  the  particle  will  describe 
the  same  path  as  before,  and  we  may  treat  the  problem  as 
if  the  particle  were  free  to  move  under  the  action  of  this 
system  of  forces,  and  therefore  apply  to  it  the  general  equa- 
tions of  motion  of  a  free  particle. 

189.  Kinetic  Energy  or  Vis  Viva  (Living  Force), 
and  Work. — A  particle  is  constrained  to  move  on  a  given 
smooth  plane  curve,  under  given  forces  in  the  plane  of  the 
curve,  to  determine  the  motion. 

Let  APC  be  the  curve  along  which  the  particle  is  com- 
l)elled  to  move  when  acted  upon  by  any  given  forces.  Let 
Ox  and  Oy  be  the  rectangular  axes  in  the  plane  of  the 


^i 


III! 


346 


KTNSTIC  ENERGY. 


curve,  ilie  axis  y  positive  up- 
wards, and  {x,  y)  the  place  of 
the  particle,  P,  at  the  time  t ; 
let  X,  Y,  parallel  respectively  to 
the  axes  of  x  and  y,  be  the  axial 
components  of  the  forces,  the 
mass  of  the  particle  being  m  ; 
let  R  bo  the  pressure  between 
the  cuvve  and  particle,  which 
acta  in  the  norma.1  to  the  curve,  since  it  is  smooth.  Then 
the  equations  of  motion  are 


FI«.83 


(IP  as 


(1) 


(3) 


Multiplying  (1)  and  (2)  respectively  by  dx  and  dy,  and 
atlAiag,  we  have 

„d^^y^fl  =  Xdx  +  Ydy. 

lutograting  between  the  limits  t  and  i^,  and  calling  v^  the 
initial  velocity,  we  have 


!J  ,^  -  'I  r,«  =   CiXlx  +  Ydy 


m 


The  t«rm  ^  ^  »«  called  the  vis  vivo/*,  or  Kinetic  Energy 

of  the  mass  w;  that  is,  vis  viva  or  kinetic  energy  is  a 
quantity  which  varies  as  the  product  of  the  mass  of  the 
particle  and  the  square  of  its  velocity.  There  is  particular 
advantage  in  def.ning  vis  viva,  or  kinotic  energy,  as  half 


t  IB  smooth.    Then 


;  (1) 

(3) 
y  by  dx  and  dy,  and 

J,  and  calling  v^  the 


+  rrfy 


(3) 


,  or  Kinetic  EniTgy 

kinetic  energy  is  a 

of  the  mass  of  Ihe 

There  is  purlicnlar 

notic  energy,  a«  half 

til,  p.  MB. 


KINETIC  SNSR&r> 


847 


the  product  of  the  mass  and  the  »quaro  of  its  velocity.* 
The  lirst  member,  therefore,  of  (3)  is  the  via  viva  or  kinetic 
energy  of  m  acquired  in  its  motion  from  (x,,  y^)  to  {x,  y) 
under  the  action  of  the  given  foi-ces. 

The  terms  Xdx  and  Ydy  are  the  products  of  the  axial 
compoL-nts  of  the  forces  by  the  axial  displacements  of  the 
mass  in  the  time  dt,  and  are  tlierefore,  tlie  elements  of  work 
done  by  the  accelerating  forces  X  and  Y  in  the  time  dt, 
according  to  the  definition  of  work  given  in  Art  101,  Eeni.; 
80  that  the  second  member  of  (3)  expresses  the  work  done 
by  these  forces  through  thfe  spaces  over  which  they  moved 
the  mass  in  the  timo  between  t^  and  /.  This  equation  is 
called  ihe  equation  of  kinetic  energy  and  of  work;  it  shows 
that  the  work  done  by  a  force  exerting  action  through  a 
given  distance,  is  equal  to  tl»o  increase  of  kinetic  energy 
which  has  accrued  to  the  mas«  in  its  motion  through  that 
distance. 

If  in  the  motion,  kinetic  energy  is  lost,  negative  work  is 
done  by  the  force  ;  i.  e.,  the  work  i;-  -.orcd  up  as  potential 
work  in  the  mriss  on  which  the  foicp  has  acted.  Thus,  if 
work  is  spent  on  winding  up  a  wulch,  that  work  is  stored, 
in  the  coiletl  spring,  and  is  thus  potential  and  ready  to  be 
'•estored  under  adajjted  circumstances.  Also,  if  a  weight  is 
raised  through  a  vertical  distance,  work  is  Bi)ent  ui  raising 
it,  and  that  work  may  l)e  recovered  by  lowering  the  weight 
through  tlie  same  voitieal  distance. 

Tliis  theorem,  in  its  most  general  form,  is  the  modern 
principle  of  nmservation  of  eneryy  ;  and  is  made  the  funda- 
mental theorem  of  abstract  dyiiamics  as  applied  to  natural 
philosophy. 

In  this  case  we  have  an  instance  of  spaee-intecirah,  which, 
as  wo  have  seen,  gives  uw  kinetic  energy  and  work  ;  tlio 
Holution  (»f  problems  of  kinetic  energy  and  work  will  1)6 
oxplaiuod  in  Chap.  V. 

•  Si)ni(>  wnUTH  iltifliiK  viR  Ttvk  a*  llii>  whiil<i  pcndiict  uf  the  UUM  and  tb«  iquar* 
<A  Ute  vviueltjr.    So*  MovUi'h  II%M  Uywuak*,  |>.  KM. 


'jTtyiiffli'wwrontUBffflw 


SB  ACTION  OF  TBS  COMiTRATNTNO   CUB  VS. 


Now  if  X  and  Y  are  functions  of  the  co-ordinates  x  and 
y  the  second  member  of  (3)  can  be  integrated  ;  let  it  be  the 
differential  of  some  function  of  x  and  y,  as  ^  (ar,  y).  Inte- 
grating (8)  on  this  hypothesis,  and  supposing  w  and  v^  to 
bb  the  velocities  of  the  particle  at  the  points  (x,  y)  and 
(*«>  Vit)  corresponding  to  t  and  t^,  "ve  have 


"c 


v«»)  =  <^  (^,  y)  -  ^  («o.  yo) 


(4) 


»^'i 


i-i'i 


which  shows  that  the  kinetic  energy  gained  by  the  particle 
constrained  to  move,  under  the  forces  X,  F,  along 
any  path  whatever,  from  the  point  (a-^,  y,)  to  the  point 
{x,  y),  is  entirely  independent  of  the  path  pursued,  and 
depends  only  upon  the  co-ordiniites  of  the  points  left  and 
unived  at;  the  reaction  R  does  not  appear,  which  is  clearly 
as  it  should  be,  since  it  does  no  work,  because  it  acts  in  a 
line  perpendicular  to  the  direction  of  motion. 

190.  To  Find  the  Reaction  of  the  Constraining 
Curve. — For  conveniens,  the  muss  oi  the  particle  iniiy  bo 
taken  as  unity.    Multiplying  (1)  and  (2)  of  Art.  189  by 

^  and  -V-;  subtracting  the  former  from  the  latter,  and 

solving  for  R,  we  have, 

„  _  (Pydx  —  rf»x  dy        -^dy       ydx 
""-    "     ilPda  "^      ds       '  ds 


«^-.  +  ^%  -  Y%^'^^n^)<^^  ^rLm    (1) 


in  which  p  is  the  radius  of  curvature  at  the  point  P.  The 
lu«t  two  terms  of  (1)  are  the  nornml  conipoiionts  of  the 
impressed  fonrs;  and  (liorefore,  if  the  purticlf  were  at  rej^t, 
tlioy  would  donule  I  be  whole  prcsaure  on  the  curve ;  but 


Mto 


fO   CUB  vs.      , 

co-ordiuates  x  and 
•ated  ;  let  it  be  tho 
as  ^  (x,  y).  lutc- 
posing  V  and  v,  to 
poiuts  (x,  y)  and 
ve 


(«o.  yo) 


(4) 


led  by  the  particle 
jes  X,  F,  along 
j»,)  to  tho  point 
)ath  pursued,  and 
the  points  left  and 
ir,  which  is  clearly 
icause  it  acts  in  a 
bion. 

;he  Constraining 

he  particle  nuiy  bo 
2)  of  Art  189  by 

in   the   latter,  and 


at 


:;i)  of  Art.  102    (1) 

the  point  P.  The 
•onipononts  of  the 
iirtich'  Were  at  rest, 
on  the  curve ;  but 


REACTION  OF  THE  CONSTRAWirfO    CURVE. 


349 


the  particle  being  in  motion,  there  ;.>  an  additional  pressure 

on  the  curve  expressed  by  -• 

In  the  above  reasoning  we  have  considered  the  particle  to 
be  on  the  concave  side  of  the  curve,  and  the  resultant  of  X 
and  Y  to  act  towards  the  convex  side  along  some  line  as  PF 
so  as  to  produce  pressure  against  the  curve.  If  ou  the 
contrary,  this  resultant  acts  towards  tho  concave  side,  along 
PF'  for  example,  then,  whether  the  jiartiole  be  on  tho 
concave  or  convex  side,  the  pressure  against  the  curve  will 

.3 

bo  the  diflEerence  between  -  and  tho  normal  resultant  of  X 

9 
and  Y. 

191.  To  Find  the  Point  where  the  Particle  WiU 
Leave  the  Constraining  Curve. — It  is  evident  that  at 
that  point.  R  =  0,  as  there  will  be  no  pressure  against  the 
curve.    Therefore  (1)  of  Art.  190  becomes 

-  =  Y~- X^ 
p  ds  ds 

=  F'  cos  F'PR 

if  F'  be  tho  resultant  of  X  and  Y. 

.-.    «)»=  F'p  con  F'PR 

=  2F'-i  chord  of  curvature  in  the  direction  PF'. 

Comparing  this  with  (6)  of  Art.  140,  we  see  that  the 
particle  will  have  the  curve  at  the  point  where  its  velocity  is 
such  as  timtild  be  produced  by  the  resultant  force  then  acting 
on  it.  if  continued  constant  during  its  fall  from  rest  through 
a  space  equal  to  J  of  the  cnord  of  curvature  parallel  to  that 
resultant.  (See  Tait  and  Steele's  Dynamics  of  a  Particle, 
p.  170.) 


m 


■/miiii>ua»!aiM.i»tvi « 


350 


COySTRAlNSD  UOTtOK. 


192.   Constrained  Motion  Undor  the  Action 
Gravity. — When  gravity  is  the  only  force  acting  on  thi 
particle,  the  formnlsB  are  simplified.     Taking  the  axis  of  y 
vertical  and  positive  downwards,  the  forces  become 

X=0,    and    r=  -{-^; 

and  for  the  velocity  we  have,  by  (3)  of  Art  189, 

where  y,  is  the  initial  space  corresponding  to  the  time  t^. 
"^r  the  pressure  on  the  curve  we  have,  by  (1)  of  Art.  190, 


p      ^  da 


m 


If  the  origin  be  where  the  motion  of  the  particle  begins, 
the  initial  velocity  and  space  are  zero,  and  (1)  becomes 


i«^==i?y. 


(8) 


This  shows  that  the  velocity  of  the  particle  at  any  time 
is  entirely  independent  of  the  form  of  the  cnrve  on  which 
it  moves ;  and  depends  fjolely  on  the  perpendicular  distance 
through  which  it  falls. 

193.  Motion  on  a  Circnlar  Aro  in  a  Vertical 
Plane. — Take  the  vertical  dianiete'  a?  axis  "of  y,  and  its 
lower  extremity  m  origin  ;  then  the  equation  of  the  circle  is 


3f  =  2ay~f; 


dT 


a-y       X 


dy  __  ds 


j4 


^■'  "(PP    ■        -^f^. 


a) 


'  the  AotLon 

irce  acting  on  thi 
;ing  the  axis  of  y 
B8  become 


r* 

rt  189, 

I  (1) 

g  to  the  time  t^. 
by  (1)  of  Art.  100, 

(3) 

le  particle  begins, 
I  (1)  becomes 

(3) 

tide  at  any  time 
le  cnrve  on  which 
endicalar  distance 


In  a  Vertical 

nxis'of  f/y  and  its 
ion  of  the  oirclo  is 


(1) 


1 


MOTTON  ON  A   CIXCULAR  ABC 

Let  (ife,  A)  be  the  point  K  where 
the  particle  starts  from  rest,  and  (x,  y) 
the  point  P  where  it  is  at  the  time  /. 
Then  the  particle  will  have  fallen 
through  the  height  HM  —  h  —  y, 
and  hence  from  (3)  of  Art.  492  we 
have 


351 


(8) 


Hence  the  velocity  is  a  minimum  when  y  =  h,  and  a 
maximum  when  y  =  0;  and  this  maximum  velocity  will 
carry  the  particle  through  0  to  iT'  at  the  distance  h  above 
the  horizontal  line  through  0. 

To  find  the  time  occupied  by  the  particle  in  its  descent 
from  K  to  the  lowest  point,  0,  we  have  from  (2) 


dt 


da 


Vajr  (*  -  y) 


—  ady 


Vfi9{h-y)(iay-f) 


by(i)    (3) 


the  negative  sign  being  taken  rinoe  /  is  a  decreasing  func- 
tion of  8. 

This  expression  does  not  admit  of  integration  ;  it  may  be 
reduced  to  an  elliptic  integral  of  the  first  kind,  and  tables 
are  given  of  the  approximate  values  of  the  integral  for 
given  values  of  y.* 

If,  however,  the  radius  of  the  circle  is  large,  and  the 
greatest  distance  KO,  over  which  the  particle  moves,  is 
Biurtll,  wo  may  develoiw  (3)  into  a  series  of  teruis  in  ascend- 
ing powers  of  |  ,  and  thus  find  the  integral  approximately. 

•  See  Leuondre  i  Twit*  des,  PoncU<in«  KUIplUiaei. 


nwimMiiWi 


• 


'isSuSistsosiiLWit 


363 


TBS  St  MPLS  PENDULUM. 


Let  ST  be  the  time  of  moti'm  of  the  particle  from  K\o  K', 
i.  e.,  from  y  =  h,  thiougli  y  =:  0,  to  y  =  h  again,  then  (3) 
becon\08 

r  =  -  \  /?  r~~J^--  (i  -  ^v^ 


~  V^'A.  L 


i  +  *^  +  H 


(ir--] 


</y 


V/iy  —  y* ' 


integrating  eacli  t«rm  eeparately  we  have 


^-^«4m 


W 


which  is  the  complete  expression  for  the  time  of  moving 
from  the  extreme  position  K  on  one  side  of  the  vertical  to 
the  extreme  position  K'  on  the  other;  this  is  called  an 
OBcillatiori.     (See  Price'e  Anal.  Mcchs.,  Vol.  III.,  p.  588). 
If  the  arc  le  very  small,  h  is  very  small  in  comparison 

with  a,  and  all  the  terms  containing  —  will  be  very  small, 

and  by  neglecting  them  (4)  becomes 


'■=Vi- 


(«) 


194.  The  Simple  Pendulnm.— Tri'^toad  of  8uppoBin/>: 
the  j article  to  move  on  a  curve,  wo  may  imagino  it  r,aa- 
pended  by  a  string  of  invariable  length,  or  a  thin  rod 
considered  of  no  weight,  and  moving  iu  a  vertical  plane 
ftlmnt  Mil'  point  0:  for,  whether  the  force  acting  on  the 
jmrticle  be  the  reaction  af  the  curve  or  the  tension  of  the 
string,  its  intensity  is  the  Hame,  while  its  direction,  in 
either  caae  is  along  the  uormai  Lu  the  curve. 


MM 


if. 


icle  from  K  to  K', 
:  h  again,  then  (3) 


•] 


dy 


^hy  —  y» ' 


•4/  W 

] 


(4) 


le  time  of  moving 
of  the  verticiil  to 
this  is  called  an 

"ol.  in.,  p.  588). 

nail  in  comparison 

will  be  very  small, 


(5) 

toad  of  supposing 
ly  imagine  it  ^ias- 
th,  or  a  thin  rod 
1  a  vertical  piano 
rce  acting  on  the 
the  tension  of  the 
e  its  dtrection,  in 
•ve. 


S^^^m^^^^^E^B^Sn 


RSLATION  OF  TIME,  LBNOTH,  STC. 


353 


When  the  particle  is  supposed  to  be  snspended  by  a 
thread  without  weight,  it  bocomes  what  is  termed  a  simple 
pendulum  ;  and  although  such  an  instrument  can  never  be 
jjerfectly  attained,  but  exists  only  in  theory,  yet  approxima- 
tions may  be  made  to  it  sufficiently  near  for  practical  pur- 
poses, and  by  means  of  Dynamics  we  may  reduce  the 
calculation  of  the  motion  of  such  a  pendulum  to  that  of 
the  simple  pendulum. 

If  I  is  the  length  of  the  rod,  the  time  of  an  oscillation  is 
approximately  given  by  the  formula 


=  ''v/l 


0) 


when  site  angle  <rf  oscillation  is  very  small,  i.  «.,  not  ex- 
ceeding »dout  4°;*  and  therefore,  for  all  angles  between 
this  and  aero,  the  tones  of  oscillation  of  the  same  pen- 
dulum will  not  perceptibly  differ ;  i.  e.,  in  very  mall  arcs 
the  os<;illatio7iH  may  be  regarded  as  isochronal,  or  as  all 
performed  in  the  same  time. 

195.    Relation  ol   Time,    Length,  end   Force    of 

aravity.— From  (I)  of  Art.  191,  we  have  Tx  y/lif  yis 

constant ;  Tec  -—  ii  I  is  conBtpnt;  gcc  l\t  Tn^  constant, 

yg 
that  is 

(1)  For  the  same  place  the  times  of  osdUaiion  are  as  the 
square  roots  of  the  lengths  cf  the  pendulums. 

{%)  For  the  same  pendulum  the  times  of  ountfntion  are 
inversely  as  the  square  roofs  of  the  force  of  gravity  a4 
different  places. 

•  If  the  InliUl  lnriln»iloo  Is  6',  the  seeoad  tmna  of  (4)  U  only  O.WOm  ;»»•*• 
•MOB*  Krm  la  only  O.UOOOitt. 


md 


354 


HSrOET  or  MOUNTAIN  BSTSBMINSD. 


(3)  For  tbs  same  time  the  lengths  of  pendulums  vary  as 
the  force  of  gravity. 

Hence  by  means  of  the  pendulum  the  force  of  gravity  at 
different  places  of  the  earth's  surface  may  be  determined. 
Let  L  be  the  length  of  a  pendulum  which  vibrates  seconds 
at  the  place  where  the  value  of  jr  is  to  be  found ;  then  ftom 
(1)  of  Art.  194  we  have 


1  =  •^Y -;  •'•  9  ~  "*^; 


(1) 


and  from  this  formula  g  has  been  calculated  at  many  places 
on  the  eai-tb.  Tho  method  of  determining  L  accurately 
will  bo  investigated  in  Chap.  VII. 

Cor. — If  n  bo  the  numl>er  of  vibratioacs  performed  dur- 
ing ^seconds,  and  iTthe  time  of  one  vibration, 


then 


=  ^by<l)ofArt.  194.  =  fy^f- 


(2) 


Since  gravity  decreases  according  to  a  known  law,  as  we 
ascend  above  the  earth's  surface,  the  comparison  of  flbe 
times  of  vibration  of  the  same  pendulum  on  the  top  of  a 
monnfain  and  at  its  base,  would  give  approximately  its 
height. 

106.  The  Height  of  a  Mountain  Determined  with 
the  Pendulum. — A  hcciiuiIh  jnniiuUtm  is  carried  to  the  top 
of  a  mountain  ;  required  to  find  ^h$  huighl  nf  the  mountain 
by  observing  the  chat^e  in  the  lime  of  oscillation. 

Tjet  r  be  the  radius  of  the  «f(  Ih  onnsidered  spherical ,  h 
the  height  of  the  mountain  above  tho  surface ;  I  the  length 
of  the  pendulum ;  g  and  g'  the  values  of  gravity  ou  the 
earth's  surface,  and  at  the  top  of  the  mountain  respectively. 
Then  (Art.  1T4)  we  have 


\XINMD. 

fndulmns  vary  as 

force  of  gravity  at 
ay  be  determined. 
I  vibrates  seconds 
found ;  then  from 


*L; 


(1) 


;ed  at  many  places 
niiig  L  accurately 

IS  performed  dur- 
iratioD, 


(2) 


known  law,  as  we 
omparison  of  the 
I  on  the  top  u(  a 
approximately  its 


>etermiiiecl  with 

s  carried  to  the  top 
hi  of  the  mountain 
illation. 

dt  red  spherical ,  h 
rface ;  I  the  length 
of  gravity  ou  the 
intiiiu  respectively. 


SB^HssMSssw^rfj&a^ss^caB 


mmm 


BEI6BT  or  MOUXTAIlf  DMTKRMINED. 


9 


g'~\    r    /' 


9 


a'  -       ^      • 

y  -  (r  -f  hy* 


355 


(1) 


which  is  the  force  of  gravity  at  the  top  of  the  mountain. 

Let  n  —  the  number  of  oscillations  which  the  seconds 
pendulum  at  the  top  of  the  mountain  makes  in  24  houi-s ; 

24  X  60  X  60 


then  the  time  of  osciUatioo 
(1)  of  Ai-t.  195,  we  have 

24  X  60  X  60 


Hence  from 


h 

r 


24  X  60  X  60 
n 


-1,  (since  Tr'Y/-  =  l),         (2) 


which  gives  the  height  of  the  mountain  in  terms  of  the 
radius  of  the  earth.  For  the  sake  of  an  example,  suppose 
the  pendulum  to  lose  5  seconds  1n  a  day  ;  that  is,  to  make 
'<  oscillations  less  than  it  would  make  on  the  surface  of  the 
earth. 


Then 


»  =  24  X  60  X  60  —  5 ; 


ii  in  (2)  gives 
A 


=  ('- 


24  X  00  X 


_U  X  60  X  60 
24  X  60  X  00  -  I 


-1 


h  = 


4000 


24  X  60  X  12 

=  J  mile,  nearly, 


nearly ; 


»4  X  6»)  X  12 

r  being  4000  miles  (approximately). 

197.  The  Depth  of  a  Mine  Oetenniued  by  Ob- 
serving the  Change  of  Oscillation  in  a  Seconds 
Pendulum.— [jf  I  r  ho  the  radius  of  the  earth  ae  in  the 


356 


CENTRIPETAL  FORCE. 


last  case  ;  h  the  depth  of  the  mine  ;  g  and  g'  the  values  of 
'  gravity  on  the  earth's  surface  and  at  the  bottom  of  tlio 
mine.    Then  (Art.  171)  we  have 


g'       r~h 


(1) 


Let  n  =  the  number  of  oscillations  which  the  S'econds 
pendulum  at  the  bottom  of  the  mine  makes  in  24  hours. 


Then 


24  X  60  X  60 
n 


_        /      Ir 


r  —  h' 


r  ~  V24  X  60  X  60/ ' 


from  which  h  can  be  found.    If,  as  before,  the  pendulum 
loses  5  seconds  a  day,  we  have 


*  =  '-(■ 


24 


__j y 

X  60  X  12/ 
nearly. 


12  X  60  X  12 

.  • .    h  =  ^  mile  nearly. 

(See  Price's  Anal.   Mech's,  Vol.  I,   p.  590,   also  Pratt's 
Mech's,  p.  376.) 


198.  Centripetal  and  Centrifugal  Forces. 


;iuce 


the  pressure  — ,  at  any  point,  depends  entirely  upon  tlie 

velocity  at  that  point  and  the  radius  of  curvature,  it  would 
remain  the  same  if  the  forces  X  and  Y  wer^  *K)th  zero,  in 
which  case  it  would  be  the  whole  norma    pressure,  B, 


I  g'  the  values  of 
lie  bottom  of  tlu; 


(1) 

hich  the  seconds 
es  in  34  hours. 


-h) 


re,  the  penduhim 


3    _, 


rJ 


590,   also  Pratt's 

Forces. — Face 

jntirely  upon  tlie 

rvfjture,  it  would 
yer«  fwth  zero,  in 
mil;  pressure,  /?. 


,,«,,  r.'»>mmimmmKgmmtmmmKt&iKKmm^ 


CBN'^    rrVQAL  FORCS. 


867 


against  the  curve.  It  is  easily  seen,  therefore,  that  this 
pressure  arises  entirely  from  the  inertia  of  the  moving 
particle,  i.  e.,  from  it-  tendency  at  any  point,  to  move  in 
the  direction  of  a  tangent ;  and  this  tendency  to  motion 
along  the  tangent  uecessurily  causo--  it  to  exert  a  pre  iire 
against  the  deflecting  curve,  and  wluJi  requires  the  curve 

4 

to  oppose  the  resistance  -•    Hence,  since  tl-    particle  if 

left  to  itself,  or  if  left  to  the  action  of  a  force  uong  the  tan- 
gent, would,  by  the  law  of  inertia,  continue  to  niovi'  along 

that  tangent,  -  is  the  effect  of  the  force  which  deflects  the 

particle  from  its  otherwise  rectilinear  path,  and  draws  it 
towards  the  centre  of  curvature.  This  force  is  called  the 
Centripetal  Force,  which,  therefore,  may  be  defined  to  be 
the  force  which  deflects  a  particle  from  its  otherwise  recti- 
linear path.  The  efjual  and  opposite  reaction  exerted  away 
from  the  centre  is  called  thi'  Centrifugal  Force,  which  may 
be  defined  to  be  the  resistance  which  the  inertia  of  a  particle 
in  motion  opposes  to  whatever  deflects  it  from  its  rectilinear 
path.  Centri[)etal  and  centrifugal  are  tlei-efore  the  same 
quantity  under  different  aspects.  The  action  of  the  former 
is  towards  the  centre  of  curvature,  while  that  of  the  latter 
wfron  the  centre  of  curvature.  The  two  are  called  central 
forces.  They  determine  the  direction  of  motion  of  the  par- 
ticle but  do  not  affect  the  velocity,  since  they  aet  continu- 
ally at  right  angles  to  its  puth.  If  a  particle,  attached  to  a 
string,  be  whirled  about  a  centre,  the  intensity  of  these 
central  forces  is  measured  by  the  tension  of  the  string.  If 
the  string  be  cut,  the  particle  will  move  along  a  tangent  to 
the  curvo  with  unchanged  velocity. 

Cob.  1.— If  m  be  the  mass  moving  with  velocity  v,  its 
centrifugal  force  is  »i  -•     If  w  bo  the  angular  velocity 


^f^i^ifv^"  ,'WrTr^ 


'iWS'^J--- 


*#«**i«»(i»rtt«|Sft^gw^i. 


MiHm 


IMAGE  EVALUATION 
TEST  TARGET  (MT-3) 


A^^. 


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y. 


«(S 


<n . 


1.0 


I.I 


iU 


ii 


25 
2.2 

2.0 


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Photographic 

Sdences 
Corporation 


33  WIST  MAIN  STRCIT 

WitSTM.N.Y.  MStO 

(716)  17:4503 


%» 

^ 


4, 


CIHM/ICMH 

Microfiche 

Series. 


CIHM/iCIVIH 
Collection  de 
microfiches. 


Canadian  Inttituta  for  Historical  IMicroraproductiont  /  Cnstitut  Canadian  dm  microraproductiont  hittoriquas 


■m' 


■■  *T"^fifirftVTMi'fr'^lnfimitTWr 


898 


cMNimaveAh  forcm, 


described  by  the  mdms  of  corvature,  thea  (Art.  160,  Ex.  1), 
V  =.  pu,  and  CQQfleqaently 


the  centrifugal  force  of  wt  =  mw^. 


(1) 


Cob.  8. — Let  m  move  in  a  cinjle  with  a  constant  velocity, 
v\  lot  a  =  the  radius  of  the  circle,  and  T  the  time  of  a 
complete  revolution ;  then  drra  :=  vT'y 


4rr*fl 


the  oentriftagal  force  of  m  =  «•  -^  ; 


w 


and  thus  the  centrifugal  force  i%  a  circle  varies  directly  as 
the  radius  of  tlie  circle,  and  inversely  as  the  square  uf  the 
periodic  time. 

Cob.  3.— 11'  fa  irsoves  in  the  ci:K!le  with   a   constant 
angular  velocity,  w,  then  (Art.  160,  Ex.  1),  t*  =  ou ; 


the  centrifugal  force  of  «i  =  tn «- *a ; 


(8) 


and  therefore  varies  directly  as  the  radius  of  the  circle. 

Thus  if  a  particle  of  mass  m  is  fastened  by  a  string  of 
length  a  to  a  point  in  a  horizontal  plane,  and  describeb  a 
circle  in  the  plane  about  the  given  point  as  centre,  the  cen- 
trifugal forco  produces  a  tension  of  the  string,  and  if  u  is 
the  constant  angular  velocity,  the  tension  =  tn  u^ 

199.  Th*  OMtriftigia  Foree  Kt  tii«  Bqnator.— Let 

R  demote  the  equatorial  radius  of  the  earth  =  209:88802* 
feet,  T  the  time  of  revolution  upon  its  axis  =  86164 
seconds,  and  -n  =  3.1418926.  Substituting  these  values  in 
(2)  of  Art.  198,  and  denoting  the  centrifugal  fbroe  at  the 
equator  by/,  and  the  mass  by  unity,  we  have 


4ffS/Z 
/=  ^^  =  0.11126  feet. 


d) 


*  Bnejp.  Brit.,  Art.  Qweimf. 


1  (Art.  160,  Ex.1), 


muV« 


(1) 


k  constant  velocity, 
T  the  time  of  a 


(2) 


( varies  directly  as 
the  square  uf  the 


with   a   constant 


m  itra  \ 


(3) 


8  of  the  circle. 
sned  by  a  string  of 
le,  and  describes  a 
as  centre,  the  cen- 
string,  and  if  u  is 
1  =  m  <.<*a, 

!•  Bqwitor.— Let 

arth  =  2093680S* 
its  axis  =  88164 
ing  these  valaes  in 
Ifugal  fbroe  at  the 
have 


ct. 


(1) 


Hi 


nn 


OMiTTBiruaAL  roBom, 


86d 


The  force  «rf  gravity  at  the  equator  has  been  found  to  be 
32.09023 ;  if  this  force  were  Eot  dinainidbed  by  the  cen- 
tiifogal  force;  t.  «.,  if  the  earth  d.l  not  revolve  on  its 
axis  the  force  of  gravity  at  the  equator  would  be 

G  =  32.09022  +  0.11126  =  32.20148  feet.         (2) 

To  determine  the  relation  between  the  oentriftigal  force 
and  the  force  of  gravity,  we  divide  (1)  by  (2)  which  gives 


l_  0.11126  _    1^  J 


(8) 


N 

C  /(? 


that  IS,  tU  centrifugal  force  at  the  equator  is  gijf  of  that 
which  the  force  of  gravity  at  the  equator  would  be  if  the 
earth  did  not  rotate. 

200.  CeiitiiltagidFonMatZMffir- 
ent  LatitadM  on  tlM  Barth.— I  at 

P  be  any  particle  on  the  earch's  surface 
describing  a  circumference  about  the 
axis,  *.V5,  with  tlie  radius  PL.  Let 
<l>  z=  AOP  =  the  latitude  of  P;  J? 
the  radius,  AC,  of  the  earth ;  and  I^ 
the  radius  PD  of  the  parallel  of  lati- 
tude passing  through  P.    Th«m  we  have 

i2'  :^  i2  cos  <t>. 

Let  the  centrifagal  force  at  tha  point  P,  whidi  is  exerted 
in  tho  direction  of  the  radius  DP,  be  repwier.ted  by  the 
lino  PB.  Resolve  this  into  the  two  components  Pf,  act- 
ing along  the  tangent,  and  PE,  acting  along  the  normal. 
Then  by  (2)  ol  Art  198  we  have 


s 
n|.« 


(1) 


,  by  (1). 


w 


360 


CSITTRtmjftAL  PORCJS. 


Hence,  the  centrifugal  force  at  any  point  on  (he  earth's 
surface  varies  directly  as  the  cosine  of  the  latitude  of  the 
place. 

For  the  normal  component  we  have 
PB  =  PB  cos  ^ 

i-n*R  co8»  0  .      ... 
= jiT—  ^7  (2) 

=  /  co8»  ^,  by  (1)  of  Art  197.  (3) 

Honce^  the  component  of  the  centrifugal  force  which  directly 
opposes  the  force  of  gravity,  at  any  point  on  the  earth's  sur- 
face,  is  equal  to  the  centrifugal  force  at  the  equator,  mul- 
tiplied by  the  square  of  the  cosine  of  the  latitude  of  the 
place. 


Also 


PF  =  PB  Bin  0 

—  ^^^  **'"  0  cos  ^ 


.  by  (2) 


.kl  . 


=  J  sin  20.  by  (1)  of  Art.  197  ;  (4) 

that  is,  the  component  of  the  centrifugal  force  which  tends 
to  draw  particles  from  any  parallel  of  laltfnde,  P,  towards 
the  equator,  and  to  cause  the  earth  to  assume  the  figure 
of  an  oblate  spheroid,  varies  as  the  sine  of  twice  the 
latitude. 

The  preceding  calculation  is  made  on  the  hypothesis  that 
the  earth  is  a  perfect  sphere,  whereas  it  is  an  oblate 
spheroid ;  and  the  attraction  of  the  eaith  on  particles  at 
its  surface  decreases  as  we  pass  from  the  poles  to  the 
equator.      The   t>endalum    furnishes  the   most   accurate 


m 


m'nt  on  (he  earth's 
'  the  latitude  of  the 


by  (2) 

(1)  of  Alt  197.   (3) 

force  which  directly 

!  on  the  earth's  sur- 

it  the  equator,  mul- 

the  latitude  of  the 


(l)of  Art.  197;(4) 

I  force  which  tends 

lattinde,  P,  towards 

}  assume  the  figure 

sine  of  twice   ike 


the  hypothesis  that 
as  it  is  an  oblate 
arth  on  particles  at 
1  the  poles  to  the 
the   most   accnrate 


TBJl  CONICAL  PENDULUX. 


861 


na-N 


method  of  determining  the  force  of  gravity  at  different 
places  on  the  earth's  surface. 

201.  The  Conical  Pendn- 
lum.— The  Gtovemor. — Suppose 
a  particle,  P,  of  mass  m,  to  be  at- 
tached to  one  end  of  a  string  of 
length  /,  the  other  end  of  which  is 
fixed  at  A.  The  particle  is  made 
to  describe  a  horizontal  circle  of 
radius  PO,  with  uniform  velocity 
rouml  the  vertical  axis  A0,&o  that 
it  makes  n  revolutions  per  second. 
It  is  required  to  find  the  inclina- 
tion, 0,  of  the  string  to  the  vertical, 
and  the  tension  of  *he  string. 

The  velocity  of  P  in  feet  per  second  =  2rrn  •  OP  =  2ir«  I 
sin  e.  The  forces  acting  upon  it  are  the  tension,  7,  of  the 
string,  the  weight  m,  of  the  particle,  and  the  centrifugal 

force,  m  — ^j— ^ —  (Art  198).    Hence  resolving,  we  have 

for  horizontal  forces,     T*  sin  (9  =  »j.  4n»»»»  /  sin  e ;  (1) 

for  vertical  forces,        iTcos©  =  mg.  (%) 

From  (1)                    7'=m.4r.V?,  (3) 

which  in  (2)  givea 

whore  7*  and  9  ere  completely  determined. 

If  the  string  be  replaced  by  u  rigid  rod,  which  can  turn 
about  ^  in  a  ball  and  socket  joint,  the  instrument  is  called 
a  conical  pendulum,  and  occurs  in  the  governor  of  the 
steam-engine. 
16 


d6d 


MXAMrXtMB. 


EXAMPLES. 

1.  If  the  length  of  the  seconds  pendalam  be  39.1393 
inches  in  London,  find  the  valne  of  jr  to  three  places  of 
decimals.  Ana.  82.191  feet. 

2.  In  what  time  wiU  a  pendnlum  nbratc  whose  length  is 
15  inches  ?  Ans.  0.63  sec.  nearly. 

3.  In  T.hat  time  will  a  pendnlnm  vibrate,  whose  length  is 
double  that  of  a  seconds  pendulum  P         Ang.  1.41  sees. 

4.  How  many  vibrations  will  a  pendalam  3  feet  long 
make  in  a  minute  ?  Ant.  62.65. 

5.  A  pendulum  which  beats  seconds,  is  taken  to  the  top 
of  a  mountain  one  mile  high ;  it  is  required  to  find  the 
number  of  seconds  which  it  wiP  lose  in  12  hours,  allowing 
the  radius  of  the  earth  to  be  4000  miles.    Ans.  10.8  sees. 

6.  What  is  the  length  of  a  pendulum  to  beat  seconds  at 
the  place  where  a  body  falls  16^  ft  in  The  first  second  7 

Ann.  39.11  ins.  r early. 

7.  If  39.11  ins.  be  taken  as  the  length  of  the  seconds 
pendulum,  how  long  must  a  pendulum  be  to  beat  10  times 
in  a  minute  ?  Ans.  117^  ft 

8.  A  particle  slides  down  the  arc  of  a  circle  to  the 
lowest  point ;  find  the  velocity  at  the  lowest  point,  if  the 
angle  described  round  the  centre  is  60°.  Ana.  Vgr. 

9.  A  pendulum  which  oscillates  in  a  second  at  one  place, 
is  carried  to  another  place  where  it  makes  120  more  oscil- 
lations in  a  day;  compare  the  force  of  gravity  at  the  latter 
place  with  that  at  the  former.  Ana,  (|fH)** 

10.  Find  the  number  of  vibrations,  n,  which  a  pendulum 
will  gain  in  Jf  seconds  by  shortening  the  length  of  the 
peudulum. 


m 


ilam  1)6  39.1393 
)  three  places  of 
1.  32.191  feeL 

whose  length  is 
63  sec.  nearly. 

,  whoso  length  is 
[ng.  1.41  sees. 

urn  3  feet  long 
Ans.  62.66. 

aken  to  the  top 
ired  to  find  the 
honre,  allowing 
Ins.  10.8  sees. 

beat  seconds  at 
first  Hecond  ? 
II  ins,  r«arly. 

I  of  the  seconds 
bo  beat  10  times 
Ans.  m^it 

a  circle  to  the 

est  point,  if  the 

Ana.  Vffr. 

•nd  at  one  place, 

120  more  oscil- 

ity  at  the  latter 

^»*.  (IIH)'.   _ 

ioh  a  pendulnm 
le  length  of  the 


Let  the  length,  i,  be  decreased  by  a  small  qnantity, 
1 1,  and  let  n  bo  increased  by  »( ;  then  from  (2)  of  Art  196 

we  get  

which,  divided  by  (2)  of  Art  195,  gives 

Ilenoe  «,  =  — 1. 

11.  If  a  pendulnm  be  45  inches  long,  ho«r  many  vibra- 
tions will  it  gain  in  one  day  if  the  bob*  be  screwed  up  one 
turn,  the  screw  having  32  threads  to  the  inch  P 

Ans.  30. 

13.  If  a  clock  loses  two  minutes  a  day,  how  many  turns 
to  the  right  hand  must  we  give  the  nut  in  order  to  correct 
its  error,  supposing  the  screw  to  have  50  threads  to  the 
inch?  •  Ans,  5*4  tarns. 

13.  A  mean  solar  day  contains  24  hours,  3  minates, 
56 '5  seconds,  sidereal  time;  calculat«d  the  length  of  the 
pendulum  of  a  clock  beating  sidereal  seconds  in  London. 
See  Ex.  1.  Ans.  38-925  inches. 

14.  A  heavy  ball,  scspended  by  a  fine  wire,  vibrates  in  a 
small  arc ;  48  vibrations  are  counted  in  3  minutes.  Cal- 
culate the  length  of  the  wire.  Ans.  45 -87  feet 

15.  The  height  of  the  cupola  of  St  Paul's,  above  the 
6oor,  is  340  ft;  calculate  the  number  of  vibmtions  a  heavy 
body  would  make  in  half  an  hour,  if  suspended  from  the 
dome  by  a  fine  wire  which  reaches  to  within  6  inches  of 
the  floor,  Ans.  176-4. 

•  The  kwer  extremity  ot  the  pendnhua. 


'I :!, 

.■I    :, 


wm 


16.  A  seconds  pendulnm  is  carried  to  the  top  of  a 
mountain  m  milea  high  ;  aasuming  that  the  force  of 
gravity  varies  inversely  as  the  square  of  the  distance  from 
the  centre  of  the  earth,  find  the  time  of  an  oscillation. 


/4000  +  m\ 


17.  Prove  that  the  lengths  of  pendulums  vibrating  dar- 
ing the  same  time  at  the  same  place  are  inversely  as  the 
squares  of  the  number  of  oscillations. 

18.  In  a  series  of  experiments  made  at  Harton  coal-pit,  a 

pendulum  which   beat  seconds  at  the  surface,  gained  2| 

beats  in  a  day  at  a  depth  of  12C0  ft. ;  if  g  and  g'  be  tho 

force  of  gravity  at  the  surface  and  at  the  depth  mentioned, 

show  that 

9'-9  _      1 

• "-  itto'o- 

19.  A  pendulum  is  found  to  make  640  vibrations  at  the 
equator  in  the  same  time  that  it  makes  641  at  Greenwich; 
if  a  string  hanging  vertically  can  just  sustain  80  lbs.  at 
Greenwich,  how  many  lbs.  can  the  same  string  sustain  at 
tho  equator  ?  Am.  80J  lbs.  about. 

20.  Find  the  time  of  descent  of  a  particle  down  the  arc 
of  a  cycloid,  the  axis  of  the  cycloid  being  vertical  and  vertex 
downward ;  and  show  that  the  time  of  descent  to  the  lowest 
point  is  the  same  whatever  point  of  the  curve  the  particle 
starts  from.  /^ 

\  9 


21.  If  in  Ex.  20  the  particle  begins  to  move  from  the 
extremity  of  the  baae  of  the  cycloid  find  the  prtssure  at  the 
lowest  point  of  the  curve. 

Ans.  2^;  i.  e.,  the  pressure  ia  twice  the  weight  of  tho 
particle. 


to  the  top  of  a 
bat  the   force    of 
the  distance  from 
m  oscillation. 
tOOQ  +  m\ 
4000     ) 


Bees. 


ms  vibrating  dnr- 
ft  inversely  as  the 


Harton  coal-pit,  a 
iirface,  gained  2| 
if  g  and  g'  be  the 
depth  mentioned, 


Tibrations  at  the 

541  at  Greenwich ; 

sustain  80  lbs.  at 

string  snetain  at 

80^  lbs.  about. 

tide  down  the  arc 
irertical  and  vertex 
Micnt  to  the  lowest 
curve  the  particle 


Am. 


Wy 


to  move  from  the 
he  pressure  at  the 


he  weight 


mmmmsmmmmmmm 


HXAMPLSa. 


d65 


%%.  Find  the  prepare  on  the  lowest  point  of  the  carve 
in  Art  193,  (1)  when  the  particle  starts  from  rest  at  the 
highest  point.  A,  (Fig.  84),  (ii)  when  it  starts  from  rest  at 
the  point  B. 

Ana.  (1)  5^;  (2)  3^;  i.e.,  (1)  the  pressure  is  five  times 
the  weight  of  the  particle  and  (2)  it  is  three  times  the 
weight  of  the  particle. 

23.  In  the  simple  pendulum  find  the  point  at  which  the 
tension  on  the  string  is  the  same  as  when  the  particle 
bangs  at  rest 

Am.  y  =  f.'.,  where  h  is  the  height  from  which  the 
pendulum  has  fallen. 

24.  If  a  particle  be  compelled  to  move  in  a  circle  with  a 
velocity  of  300  yards  per  minute,  the  radius  of  the  circle 
being  16  ft,  find  the  centrifugal  forco. 

Ana.  14' 06  ft.  per  sec. 

25.  If  a  body,  weighing  17  tons,  move  on  the  circum- 
ference of  a  circlo,  whose  radius  is  1110  ft.,  with  a  velocity 
of  16  ft  per  sec.,  find  the  centrifugal  force  in  tons  (take 
g  =  32-1948).  Am.  0-1217  ton. 

26.  If  a  l)ody,  weighing  1000  lbs.,  be  constrained  to  move 
in  a  circle,  whose  radius  is  100  ft,  by  means  of  a  string 
capable  of  sustaining  a  strain  not  exceeding  450  lbs.,  find 
the  velocity  at  the  moment  the  string  breaks. 

Ana.  38-06  ft  per  sec. 

27.  If  a  railway  carriage,  weighing  7-21  tons,  moving  at 
the  rate  of  30  miles  per  hour,  describe  a  portion  of  a  circle 
whose  radius  is  460  yards,  find  its  centrifugal  force  in  tons. 

Am.  0-314  ton. 

28.  If  the  centrifugal  force,  in  a  circle  of  100  ft  radius, 
be  146  ft  per  sec,  find  the  periodic  time. 

Ana.  6-2  sees. 


4^1 ; 


366 


XXAMPnSA 


29.  If  the  oentrifa^l  force  be  131  oss.,  and  the  radias 
of  the  circle  100  ft.,  the  periodic  time  being  one  hour,  find 
the  weight  of  the  body.  Ans.  386-  309  tons. 

30.  Find  the  force  towards  the  centre  required  to  make 
a  body  move  uniformly  in  a  circle  whose  mdius  is  5  ft., 
with  Huch  a  velocity  as  to  complete  a  revolution  in  6  sees. 

5  ■ 


Ans. 


31.  A  stone  of  one  lb.  weight  is  whirled  round  horizon- 
tally by  a  string  two  yards  long  having  one  end  fixed ;  find 
the  time  of  revolution  when  the  tension  of  the  string  is  3  lbs. 


Ana.  2rT 


VI 


sees. 


38.  A  weight,  w,  is  placed  on  a  horieontal  bar,  OA, 
which  is  made  to  revolve  round  a  vertical  axis  at  0,  with 
the  angular  velocity  «;  it  is  required  to  determine  the 
position,  A,  of  the  weight,  when  it  is  upon  the  point  of 
sliding,  the  coefficient  of  friction  being  /. 


Am.  OA 


33,  Find  the  diminution  of  gravity  at  the  Sun's  equator 
caused  by  the  centrifugal  force,  the  radius  of  the  Sun  being 
441000  miles,  and  the  time  of  levolation  on  his  axis  being 
607  h.  48  m.  Ans.  0-  0192  ft.  per  8e<;. 

84.  Find  the  centrifugal  force  at  the  equator  of  Mercury, 
the  radius  being  1670  miles,  and  the  time  of  revolution 
24  h.  6  m.  Ans.  0-  0435  ft.  per  sec 

35.  Find  the  centrifugaJ  force  at  the  equator,  (1)  of 
Venus,  radius  being  3900  miles  and  time  of  revolution 
23  h.  21  m.,  (2)  of  Mars,  radius  being  2050  miles  and 
iwriodic  time  34  h.  37  m.,  (3)  of  Jupiter,  radius  being 
43500  miles  and  periodic  time  9  h.  56  m.,  and  (4)  of  Saturn, 
radius  being  39580  miles  and  periodic  time  10  h.  29  m. 


(8.,  and  tbe  radias 
;ing  one  hour,  find 
s.  386-309  tons. 

required  to  make 
086  rndiuB  is  5  ft., 
olution  in  6  sees. 

Ana.  -=-' 

0 

ed  round  horizon- 
ne  end  fixed ;  find 
!  the  string  is  3  lbs. 


s.  2n 


VI 


sees. 


rieontal  bar,  OA, 

cal  axis  at  0,  with 

to  determine  the 

ipou  tbe  point  of 


Ans.  OA 


the  Sun's  equator 
8  of  the  Sun  being 
1  on  his  axis  being 
Qin  ft.  per  8e<;. 

jnator  of  Mercury, 
ime  of  revolution 
0435  ft.  per  sec 

le  equator,  (1)  of 
mo  of  revolution 
r  2050  miles  and 
iter,  radius  being 
and  (4)  of  Saturn, 
le  10  h.  29  m. 


mmmimmmmmmsmmmim 


MXAMPLXa. 


867 


Ana.  (1)  0>  11504  ft.  per  sec.;  (2)  0>0544  ft  per  sec.; 
(3)  7-0907  ft  per  8oa;  (4)  6-  7924  ft  per  sea 

36.  Find  the  effect  of  centrifngal  force  in  diminishing 
gravity  in  the  latitude  of  60°.    [See  (3)  of  Art.  200). 

Ana.  0-028  ft.  per  sec. 

37.  Find  (1)  the  diminution  of  gravity  caused  by  cen- 
trifugal force,  and  (2)  the  component  which  urges  particles 
towards  the  equator,  at  the  latitude  uf  23°. 

Ans.  (1)  0-09  ft  per  sec;  (2)  0-04  ft  per  sec. 

38.  A  railway  carriage,  weighing  12  tons,  is  moving 
along  a  circle  of  radius  720  yards,  at  the  rate  of  32  miles 
an  hour;  find  the  horizontal  pressure  on  the  nils. 

Ana.  0-39  ton,  nearly. 

39.  A  railway  train  is  going  ouoothly  along  a  curve  of 
500  yards  radius  at  the  rate  of  30  miles  an  hour ;  find  at 
what  angle  a  plumb-line  hanging  in  one  of  the  carriages 
will  be  inclined  to  the  vertical  Am.  2°  14'  nearly. 

40.  The  attractive  force  of  a  mountain  horizontally  is/, 
and  the  force  of  gravity  is  ^;  show  that  the  time  of  vibra- 

tion  of  a  pendulum  will  be  ww   j-    y^; 

of  the  pendulum. 

41 1  In  motion  of  a  particle  down  a  cycloid  prove  that  the 
vertical  Telocity  is  greatest  when  it  has  completed  half  its 
vertical  descent 

42.  When  a  partide  falls  from  the  highest  to  the  lowest 
point  of  a  cycloid  show  that  it  describes  half  the  path  in 
two-thirds  of  the  time. 

43.  A  railway  train  is  moving  smoothly  along  a  curve  at 
the  rate  of  60  miles  an  hoar,  and  in  one  of  the  carriages  a 
pendulum,  which  would  ordinarily  oscillate  seconds,  is 
observed  to  oscillate  121  times  in  two  minutes.  Show  that 
the  radiOB  of  tiia  ourve  is  very  nearly  a  qiiarter  of  a  mile. 


a  being  the  length 


368 


EXAMPLES. 


44.  One  end  of  a  string  is  fixed ;  to  the  other  end  a 
particle  is  attached  which  describes  a  horissontal  circle  with 
uniform  velocity  so  that  the  string  is  always  inclined  at  an 
angle  of  60"  to  the  vertical ;  show  that  the  velocity  of  the 
particle  is  that  which  would  be  acquired  in  falling  freely 
from  rest  through  a  space  equal  to  three-fourths  of  the 
length  of  the  string. 

■15.  The  horizontal  attraction  of  a  mountain  on  a  particle 
at  a  cciiain  place  is  such  as  would  produce  in  it  an  accelera- 
tion denotfld  by  "•     Show  that  a  seconds  pendulum  at  that 

,         .,,      .    21600.     ,   .       ,  , 

place  will  gam  — j—  beats  ma  da>,  very  nearly. 

46.  In  Art  201,  suppose  I  equal  to  2  ft.  and  m  to  be  20 
lbs.,  and  that  the  system  makes  10  revolutions  per  sec.,  and 
^  =  32;  find  e  and  T. 

Ans.  e  —  co8-»  — ^;  T  =  SOOtt* pounds. 

47.  A  tube,  bent  into  the  form  of  a  plane  curve,  revolves 
with  a  given  angular  velocity,  about  its  vertical  axis ;  it  is 
required  to  determine  the  form  of  the  tube,  when  a  heavy 
particle  placed  in  it  remains  at  rest  in  all  parts  of  the 
tube. 

(Take  the  vertical  axis  for  the  axis  of  y,  and  the  axis  of  x 
horizontal,  and  let  <o  =  the  constant  angular  velocity). 
Ans.  aj^uS  =  'igy,  if  a;  =  0  when  y  =  0,  t.  c,  the  curve 
is  a  parabola  whose  axis  is  vertical  and  vertex  downwards. 

48.  A  particle  moves  in  a  smooth  straight  tube  which 
revolves  with  constant  angular  velocity  round  a  vertical 
axis  to  which  it  is  perpendicular,  to  determine  the  curve 
traced  by  the  particle. 

Let  <i)  =  the  constant  angular  velocity ;  and  (r,  0)  the 
position  of  the  particle  at  -the  time  /,  and  lot  r  =:  a  when 


the  other  end  a 
lontal  circle  with 
^s  inclined  at  an 
le  velocity  of  the 
in  falling  freely 
:e-fourtha  of  the 


tain  on  a  particle 
in  it  an  accelera- 

lendulnm  at  that 
early. 

and  nt  to  be  20 
ions  per  sec.,  and 

SOOttS  pounds. 

le  curve,  revolves 
ertical  axis ;  it  is 
},  when  a  heavy 
all  parts  of  the 

and  the  axis  of  x 
Qgular  velocity). 
t.  c,  the  curve 
bex  downwards. 

ight  tube  which 
round  a  vertical 
rmine  the  curve 

;  and  (r,  0)  the 
.  lot  r  =s  a  when 


M''t(WJSlflfiWltfl!rtfll|JWlWffe:J'v 


SXAMPLSa. 


369 


/  =  0.     Then  since  the  motion  of   the  ^mrticle  ia  due 
entirely  to  the  centrifhgal  force,  we  have 


dr 


if  ^  =  0,  when  r  =  o.    Hence  we  have 


CHAPTER    IV.    . 


IMPACT. 


202.  An  Xsppnlsive  Force.— Hitherto  wc  have  con- 
sidered force  only  as  continuous,  i.  e.,  as  acting  through  a 
definite  and  finite  portion  of  time,  and  producing  a  finite 
chango  of  velocity  in  that  time.  Guch  a  force  is  measured 
at  any  instant  by  the  mass  on  which  it  acts  multiplied  by 
the  acceleration  which  it  causca  If  a  particle  of  mass  m  be 
moving  with  a  velocity  v,  and  be  reta;ded  by  a  constant 
ibrce  which  brings  it  to  rest  in  the  time  /,  then  the  measure 

of  this  force  is  -r-  (Art.  20).  Now  suppose  the  lime  t  dur- 
ing which  the  particli  is  brought  to  rest  to  be  made  very 
ginall  then  tl»o  force  required  to  bring  it  to  rest  must  bo 
▼cry  large  ;  rod  if  wo  suppose  t  so  smal!  that  we  are  unable 
to  measure  it,  then  the  force  becomes  so  great  that  we  are 
nnablo  to  obtain  its  measure.  A  typical  cnse  is  the  blow  of 
a  hammer.  Here  the  time  during  which  there  ir  contact  is 
apparently  iinlnitcsimal,  certainly  too  small  to  be  measured 
by  any  ordinary  methwle ;  yet  the  effect  produced  is  con- 
Bidcrable.  Similarly  wiien  a  cricket  ball  ia  driven  back  by 
a  blow  from  a  but,  the  original  velocity  of  the  ball  is 
destroyed  and  a  new  velocity  generated.  Also  \,hen  a  bul- 
let is  discharged  from  a  gun,  a  large  veloc'ty  is  generattid 
in  an  extremely  brief  time.  Forcee  acting  in  this  way  are 
called  impulsive  forces.  An  impulsive  force  may  thertfore 
h'  defined  to  he  a  forc^.  which  protluces  a  finite  change  cf 
motion  in  an  indefinitely  brief  time  An  Impulse  is  the 
effect  of  o  hhw. 

In  duch  cases  as  ^hese  it  is  inipossiblo  accurately  to 
determine  the  force  and  time ;    out  wo  can   dotormino 


■i 


^ 


imi 


tmm-- 


to  wo  have  con- 
octiog  thiough  a 
i-oducing  a  finite 
brcc  is  measured 
icta  multiplied  by 
^!cle  of  mass  m  be 
ed  by  a  conataut 
then  the  measure 

ie  the  time  t  dyr- 

to  be  mmle  very 
to  rest  must  bo 
lat  wo  are  unable 
great  that  we  are 
ise  is  the  blow  of 
here  ir  contact  is 
11  to  be  measured 
prodticed  is  con- 
la  driven  back  by 
ty  of  the  ball  is 
Also  \,hen  a  bul- 
c'ty  is  generattid 
[  in  this  way  are 
'ce  may  thert/ore 
I  finite  change  cf 
n  Impulse  is  the 

»lo  accurately  to 
0  can   dotormino 


IMPACT  on  COLLISrOlf. 


371 


their  product,  or  P^,  since  thia  is  merely  the  change 
in  velocity  caused  by  the  blow  (Art.  20).  Hence,  in 
the  case  of  blows,  or  impulsive  forces,  we  do  not  attempt 
to  measure  the  force  and  the  time  of  action  separately,  but 
simply  take  the  whole  momentum  producefl  or  destroyed,  as 
the  mcasnre  of  the  impulse.  Because  impulsive  forces  pro- 
duce their  effects  in  an  indefinitely  short  time  they  are 
sometimes  called  instantaneous  forces,  i.  c.,  forces  requinng 
no  time  for  thdr  ection.  But  no  such  force  exists  in 
nature;  every  force  requires  time  for  its  action.  There  is 
no  case  in  nature  in  which  a  finit«  change  of  motion  ie 
produced  in  an  infinitesimal  .-f  time  ;  for,  whenever  a 
finite  velocity  is  generated  or  destroyed,  a  finite  time  is 
occupied  in  the  process,  though  we  may  be  unable  to 
measure  it,  even  approximately. 

203.  Impact  or  CoUisioa— When  two  bodies  in  rela- 
tive motion  come  into  contact  with  each  jther,  an  impact 
or  collision  is  said  to  take  place,  and  pressure  begins  to  act 
between  them  to  prevent  any  of  their  parts  from  jointly 
occupying  the  same  space.  This  force  increases  from  zero, 
when  the  collision  begins,  np  to  a  very  large  magnitude  at 
the  instant  of  greatest  compression.  If,  as  is  always  the 
case  In  nature,  each  body  possesses  some  degree  of  eksticity, 
and  if  they  are  not  kept  together  aft«r  the  impact  by 
cohe,  ion  or  by  some  artificial  means,  the  mutual  pressure 
l)ctwe^n  them,  after  reaching  a  madmum,  will  g.-adually 
<limin.8h  to  zero.  The  whole  process  would  occupy  not 
greatly  more  or  less  than  an  hour  if  the  bo'iios  were  of  such 
dimensions  as  the  earth,  and  such  degrees  of  rigidity  m 
copper,  steel,  or  glass.  In  the  case,  however,  of  globes  of 
tliese  substances  not  exceeding  a  yard  in  diameter,  t>e 
whole  process  is  probably  finished  within  a  thousandth  of 
11  socond.* 

The  impulsive  forces  are  bo  much  more  intense  than  the 

•  TbomMD  tud  TtliV  Nkt.  Pbtl.,  p.  fti. 


SdiP 


9n 


DIBBCT  AND  CJSNTRAL  IMPACT. 


ordinary  forces,  that  daring  the  brief  time  in  which  the 
former  act^  an  ordiuar}'  force  does  not  produce  an  effect 
comparable  in  amount  with  that  produced  by  an  impulsive 
force.  For  example,  an  impulsive  force  might  generate  a 
velocity  of  1000  in  less  time  than  one-U^nth  of  a  second, 
while  gravity  in  one-tenth  of  a  second  would  generate  a 
-  elocity  of  about  three.  Hence,  in  dealing  with  the  effects 
of  impulses.  Unite  forces  need  not  be  conaidei'ed. 

204.  Direct  and  Central  Impact— When  two  bodies 
impinge  on  oach  other,  so  that  their  centres  before  impact 
are  movic^?  in  the  same  straight  line,  and  the  common  tan- 
gent at  the  point  of  contact  is  perpendicular  to  the  line  of 
motion,  the  impact  is  said  to  be  direct  and  central.  When 
these  conditions  are  not  fulfilled,  the  impact  is  said  to  be 
obligue. 

When  two  bodies  impinge  directly,  one  upon  the  other, 
the  mutual  action  between  them,  at  any  instant,  must  be 
in  the  line  joining  tbeir  centres ;  and  by  the  third  law 
(A?*t.  166),  it  must  be  equal  in  amount  on  the  two  bodies. 
Hence,  by  Law  II,  they  must  experience  equal  changes  of 
motion  iu  contrary  directions. 

We  may  consider  the  impact  as  consisting  of  two  parts ; 
during  the  first  part  the  bodies  are  coming  into  closer  con- 
tact with  each  other,  mntvally  displacing  the  particles  in 
the  vicinity  of  the  point  of  contact,  producing  a  compres- 
sion and  distortion  about  that  noint,  which  increases  till  it 
reaches  a  maximum,  when  the  molecular  reactions,  thns 
called  into  play,  are  sntfioient  to  resist  farther  compression 
and  distortion.  At  this  iusiant  it  is  evident  that  the 
points  in  contact  are  moving  with  the  same  velocity.  No 
body  in  nature  is  perfectly  itietantic ;  and  henoe,  at  the 
instant  of  greatest  compression,  the  elastic  force*  of  resti- 
tution are  brought  into  action  ;  and  during -the  second  part 
of  the  impact  the  mutual  pressure,  produced  by  the  elastic 
forcoB,  which  were  brought  into  action  by  the  compression 


«■« 


ACT. 

nc  in  which  the 
produce  an  cfFect 
by  an  impnlsive 
might  generate  a 
nth  of  a  second, 
vonid  generate  a 
%  with  the  effects 
dei-ed. 

-When  two  bodies 
es  before  impact 
the  common  tan- 
lar  to  the  Hne  of 
I  central.  When 
uct  is  said  to  be 

e  upon  the  other, 
instant,  must  be 
by  the  third  law 
I  tlie  two  bodies, 
equal  changes  of 

ng  of  two  i»rt« ; 
f  into  closer  con- 

the  particles  in 
icing  a  compres- 
b  increases  till  it 
T  reactions,  thus 
■ther  compression 
evident  th&t  the 
me  velocity.  No 
nd  henoe,  at  the 
c  fortm  of  resti- 
gthe  second  part 
led  by  the  elastic 

the  compression 


SLASTICJTy  OF  BODIMS. 


mmmsmamnmmi j. 


373 


during  the  first  part  of  the  impact,  tend  to  separate  the 
two  bodies,  and  to  restore  them  to  their  original  form. 

205.  Elasticity  of  Bodiea.-^o«Aci«at  of  Resti- 
tution. — It  appears  from  experiment  that  bodies  may  be 
compressed  in  various  degrees,  and  recover  more  or  less 
their  original  forms  after  the  compreMing  force  has  ceased ; 
this  property  is  termed  ekiHticity.  The  force  urging  the 
approach  of  bodies  is  called  the  force  of  compression  ;  the 
force  causing  the  bodies  to  separate  again  is  called  the 
force  of  reditution.  Elastic  bodies  are  such  as  regain  a 
part  or  all  of  their  original  form  when  the  compressing 
force  is  removed.  The  ratio  of  the  force  of  restitution  to 
that  of  compression  is  called  the  Coefficimt  of  JiestituHon.* 
It  has  been  found  that  this  ratio,  in  the, same  bodies,  is 
constant  whatever  may  be  their  velocities. 

When  this  ratio  is  unity  the  two  forces  ore  equal,  and  the 
body  is  said  to  lie  perfectly  elaeticj  when  the  ratio  is  aero, 
or  the  force  of  restitution  is  nothing,  the  body  is  said  to  bo 
non-elastic;  when  the  ratio  is  greater  than  zero  and  less 
than  unity,  the  body  is  said  to  be  imperfectly  elastic.  There 
are  no  bodies  either  perfectly  elastic  or  perfectly  non-elas- 
tic, all  being  more  or  less  elastic. 

In  the  cases  discussed  the  bodies  will  be  supposed  ppher- 
ical,  and  in  the  case  of  direct  impact  of  smooth  spheres  it 
is  evident  that  they  may  be  considered  as  particles,  since 
they  are  symmetrical  with  respect  to  the  line  joining  their 
centres. 

The  theory  of  the  impact  of  bodiea  ia  chiefly  due  to 
Newton,  who  found,  in  his  experiments,  that,  provided  the 
impact  is  not  so  violent  as  to  make  any  seuHiblo  iidenttition 
in  eitiier  body,  the  relative  velocity  of  separation  after  the 
impact  bears  a  ratio  to  the  relative  velocity  of  approach 
before  the  impact,  which  is  constant  for  the  same  two 

•  BoneUmea  called  CmIIIcIsd(  of  IhMtlcitr.    To<bmlar'»  Heck. .  p.  tm. 


374 


DIRECT  IMPACT  OF  I.VELASTIC  BODIES. 


bodies.  In  Newton's  experiments,  however,  the  two  bodies 
seem  always  to  have  been  formed  of  the  same  sub- 
stance. He  found  that  the  value  of  this  ratio  (the  coeffi- 
cient of  rettitution),  for  balls  of  compressed  wool  was  about 
f,  steel  about  the  same,  cork  a  little  less,  ivory  {,  glass  \\. 
The  results  of  more  i-ecent  experiments,  made  by  Mr. 
Hodgkinson,  and  recorded  in  the  Report  of  the  British 
Association  for  1834,  show  that  the  theory  may  be  received 
as  satisfactory,  with  the  exception  that  the  value  of  the 
ratio,  instead  of  being  quite  constant,  diminishes  when  the 
velocities  are  very  large. 

206.  Direct  Impact  of  XneUuitic  Bodies. — A  sphere 
of  mass  M,  moving  with  a  velocity  v,  overtakes  and  impinges 
directly  on  another  sphere  of  mass  M',  moving  in  the  same 
direction  with  velocity  v',  and  at  the  instant  of  greatest 
mutual  eomj'Tession  the  spheres  are  moving  with  a  com)non 
velocity  V.  Determine  the  motion  after  impact,  and  the 
impulse  during  the  compression. 

Jjst  R  denote  the  impulse  during  the  compression,  which 
acts  en  each  body  in  opposite  directions  ;  and  let  us  sup- 
pose the  bodies  to  be  moving  fh)m  left  to  right.  Then, 
since  the  impulse  is  measured  by  the  amount  of  momentum 
gained  by  one  of  the  impinging  bodies  or  lost  by  the  other 
(Art  202),  we  have 

Momentum  lost  hy  M  =  M{v  —  V)  -  R,  (1) 

«      gained  by  M'  =  M' (V  -  v')  =  R,  (2) 

.-.    M{v-  V)  =  M'iV-v').  (3) 

Solving  (3)  for  V  wo  get 


F  = 


which  in  (1)  or  (2)  givea 


Mv  +  M'v' 

M+sr* 


(*) 


BODtSS, 

r,  the  two  bodies 
the  same  sub- 
J  ratio  (the  coeffi- 
l  wool  was  about 
'^orj  I.  glass  +f 
I,  made  by  Mr, 
t  of  the  British 
may  be  received 
he  value  of  the 
iiishes  when  the 


dies. — A  sphere 
tea  and  impingea 
ing  in  the  same 
iant  of  greatest 
'  with  a  conAnon 
impact,  and  the 

ipresBion,  which 
md  let  us  sup- 
o  right.  Then, 
It  of  momentum 
)8t  by  the  other 


fti^MHiwWius^aiLiilwg'MW 


DIBBCT  IMPACT  OF  mSLASTiC  BODIES, 


„  _  MM'jV  -  V) 


Hence  the  common  velocities  of  the  two  bodies  after  impact 
is  equal  to  the  algebraic  sum  of  their  momenta,  divided  by 
the  sum  of  their  masses,  and  also,  from  (3),  t/is  whole 
momentum  after  impact  is  equal  to  the  sum  of  the  momenta 
before. 

Cob.  1. — Had  the  balls  been  moving  in  opposite  direc- 
tions, for  example  had  M'  been  moving  from  right  to  left, 
v'  would  httve  been  negative,  in  which  case  we  would  have 


^       Mv  -  M'v'  -     _       MM'  (r  +  t;') 


(6) 


)  =  Ii, 

(1) 

v')  =  B, 

(2) 

V'). 

(8) 
(♦) 

1 

1 

■ 

From  the  first  of  these  it  follows  that  both  balls  will  be 
reduced  to  rest  if 

Mv  =  M'v'; 

that  is,  if  before  impact  they  have  equal  and  opposite 
momenta. 

Cob.  2. — If  M'  is  at  rest  before  impact,  v'  =  0,  and  (4) 
becomes 

If  the  masses  are  equal  we  have  from  (4)  and  (6) 

V 


*   ~       2     ' 


or 


—  V 


(8) 


according  as  they  move  in  the  same  or  in  opposite  direc- 
tions. 

207.  Direct  Impact  of  Ela«1ac  Bodies.— When  thei 
balls  are  elastic  the  problem  is  the  same,  up  to  the  instant 
of  greatest  compi-ession,  as  if  they  were  inelastic ;  but  at 


;  i  ' 


•■  I 


mm 


376 


VIBMCT  IMPACT  OF  tHELASTtC  BODUta. 


tbw  initant,  the  force  of  restitation,  or  th«t  tendency  which 
elMtio  bodies  have  to  regain  their  ranginal  form,  begins  to 
throw  <me  ball  forward  with  the  same  momentum  that  it 
throws  the  other  back,  and  this  matnal  pressure  is  propor- 
tional to  R  (Art.  205). 

I«t  s  be  the  coefficient  of  restitation ;  then  daring  the 
second  part  of  the  impact,  an  impulse,  eR,  acts  on  each 
ball  in  the  same  direction  respectirely  as  R  acted  daring 
the  compression.  Let  v,  and  v,'  be  the  velocities  of  the 
balls  JTand  M'  when  they  are  finally  separated.  Then  we 
have,  as  before. 

Momentum  lost  hy  M r=  M(V —v^)  =  eR,      (l) 
*       gained  by  M'  =  M'  (r/  -  F)  =  eR.    (2) 

From  (1)  we  have 

eR  • 


V,  =  V- 


M 


i.j 


Mvj-  M'v 


M+M'  ^^  -  "^ 

by  (4)  and  (6)  of  Art.  306, 


*'  -  mTjt  <^  +  '>  <"  -"')• 


p  Similarly  from  (8)  wo  have 

M 


*'•'=•»''  ^ft:f'(i  +  ')(*'-»^)5 


(3) 


(*) 


which  are  the  velocities  of  the  balls  when  finally  eeparated 

These  results  may  be  more  easily  obtained  by  the  oon- 
sidorution  that  the  whole  impulse  is  (1  +  e)  R;  for  this 
gives  at  once  the  whole  momentum  lost  by  IT  or  gained  by 
M'  dnring  oomprMnon  and  restitution  as  follows : 


M(v  -  V,)  =  (1  -f-  «)  J?, 


(6) 


■\?ia»»-'ijMWW«Bi*w«wi»»»ww»'wiiiiiiw'»iMnw^ 


tt  tendency  which 
il  fonn,  begins  to 
lomentum  that  it 
reflsure  is  propor- 

;  then  daring  the 
eR,  acta  on  each 
18  B  acted  daring 
i  velocities  of  the 
irated.    Then  we 


■  v.)  =  en,      (1) 
-  F)  =  eB.    (2) 


-V) 

d  (5)  of  Art  306, 

"').  (3) 


t^h 


(*) 


uiUy  eeparakd. 
ined  by  the  oon- 
+•  «)  -S ;  for  this 
Mot  gained  by 
Pollows : 


and 


DISKCT  IMPJLOT  Of  IttBhASTIO  BODTKS.  377 

M'{v,'^v')=z{l  +  e)Jt.  (6) 


Substituting  in  (5)  and  (6)  the  value  of  It  from  (5)  of  Art. 
206,  we  have  the  values  of  v  and  v,'  immediately. 

Cob.  1. — If  the  balls  are  moving  in  opposite  directions* 
«'  becomes  negative.  If  the  balls  are  non-elastic,  e  =  0, 
and  (3)  rad  (4)  reduce  to  (4)  of  Art  206,  as  they  should. 

Cob.  2.— If  the  balls  are  perfectly  elaatie, «  =  1,  and  (3) 
and  (4)  become 


^^'  =  V+-J^,(v-v'). 


(8) 


M+  M 

Cob.  3,— Subtracting  (4)  from  (3)  and  reducing,  we  get 
»,  —  »,'  =  r  ~  t>—  (1  +  e)  {v  r-  v'), 


Hence,  tks  relative  velocity  after  impact  ««  —  e  times  the 
relative  velocity  before  impact. 

Cob.  4.— Multiplying  (8)  and  (4)  by  M  and  M',  respeet- 
ively,  and  adding,  we  get 


Mv,  +  M'v,'  =r  ift?  ^.  M'v'. 


(10) 


Hence,  as  in  Art.  206,  the  algebraic  turn  of  the  mometUa 
after  impact  ie  the  same  at  before;  i.  e.,  there  i»  no  mo- 
mentum lost,  which  of  conne  is  a  direet  oonseqnence  of  the 
third  law  of  motion  (Art  169). 

Cob.  6. — Suppose  «*  =  0,  so  ^t  the  body  ot  mass  M, 
moving  witii  velocity  v,  impinges  on  a.tody  of  mass  M'  at' 
rest,  th$n  (3)  and  (4)  become 


878 


LOSB  OP  KINSTW  BNXROT. 


M-eM' 


.        ,       i/  (1  +  «) 
V,    and     t»,   =    jy^jyrt'. 


(11) 


Hence  the  body  wJiich  is  struck  goes  onwards ;  and  the 
striking  body  goes  onwai-ds,  or  stops,  or  goes  backwards, 
according  as  Jif  is  gfreater  than,  equal  to,  or  less  than  eM'. 
If  M'  =  eM,  then  (11)  becomes 


V,  =  (1  —  e)  V,    and    »,'  =  v. 


(12) 


OOB.  6.— If  M  =  M  and  e  =  1 ;  that  is,  if  the  balls 
are  of  equal  mass,  and  perfectly  elastic,*  then  (7)  and  (8) 
become,  respectively, 


Vx  =  »',    and    <  =  t>; 


(13) 


that  is,  the  balls  interchange  their  velocities,  and  the 
motion  is  the  same  as  if  they  had  passed  through  one 
another  without  exerting  any  mutual  action  whatever. 

Cob.  7.— If  M'  be  inflnite,  and  v'  =  0,  we  have  the  case 
of  a  ball  impinging  directly  upon  a  fixed  surface ;  substi- 
tnting  these  values  in  (3)  it  becomes 


Vt=  —w, 


(14) 


that  is,  the  baU  rebounds  from  the  fixed  surface  unth  a  veloc- 
ity e  times  that  with  which  it  impinged. 

208.  Lou  of  Kinetic  Bnexgyf  in  the  Zmpaet  of 

Bodies.— Squaring  (9)  of  Art  207,  and  multiplying  it  by 
MM',  we  have 


MM'  (r,  -  »,')»  =  MM'  ^(v-i/y 
=  MM'{v-  v'y  -(!-«»)  MM'  (V  -  »')». 


(1) 


•  ThU  it  the  nioal  phnMoiogy,  bat  iiiialaadiiiK,  Incy.  Brit,  Vid.  XV,  Art. 
Xceh'f. 

t  8m  Art.  189. 


kmf 


<r. 

r^jgrv.       (11) 

nwards;  and  the 
'  goes  backwards, 
ar  less  thau  eJf'. 


=  V. 


(12) 


at  is,  if  the  balls 
then  (7)  and  (8) 

(13) 

locities,  and  the 
ssed  through  one 
m  whatevef. 

we  have  the  case 
I  surface;  snbsti- 

(U) 
•face  with  a  vdoe- 

the  Impact  of 

multiplying  it  by 

iV-vJ'       (1) 

By,  Brft,  Vd.  XV,  Art. 


mMimHiiyi 


Bm 


LOSS  OF  KINSTtC  ENSROT.  379 

Squaring  (10)  of  Art  207,  we  have 

(J/»,  +  M'   y  =  {Mv  +  M'v'f.  (2) 

Adding  (1)  and  (2),  we  get 

(if  +  M")  (ifv,»  +  Jf'»,'»)  =  ( Jr  -I-  M')  (Mv*  +  M'v'*) 

-  (1  -  e»)  MM'  {V  -  v'y ; 
.-.    iJfr,»  +  ii/"r,'»  =  ^Mv»+  iJfV* 

the  last  term  of  which  is  the  loss  of  kinetic  energy  by 
impact,  since  e  can  never  be  greater  than  unity.  Hence, 
there  is  always  a  loss  of  kinetic  energy  by  impact,  except 
when  e  =  1,  in  which  case  the  Iris  is  zero;  i.  «.,  when  the 
coeflBcient  of  restitution  is  unity,  no  kinetic  energy  is  lost 
When  «  =  0  the  loss  is  the  greatest,  and  equal  to 

.     ^STTM'^''-''^'' 

From  (3)  we  see  that  during  compression  kinetic  energy 
MM' 
to  the  amount  of  \  m  ^  -u'  (^  —  *')*  *^  ^<^* '   *°^  ***®" 

during  restitution,  e*  times  this  amount  is  regained. 

RiBM.— From  the  theory  of  kinetic  energy  it  appears 
that,  in  every  case  in  which  energy  is  lost  by  resistance, 
heat  is  generated;  and  from  Joule's*  investigations  we 
learn  that  the  quantity  of  heat  so  generated  is  a  perfectly 
definite  equivalent  for  the  energy  lost ;  and  also  that,  in 

•  Bee  "The  ConelatioB  eDd  Coiuemtlon  of  Force*,"  by  Heimholta,  FuwUtTt 
LieMg,  etc. ;  elM  "  Hett  M  •  Mode  of  Motion,''  by  Prof.  TyndaU.  Also  a  Stewart's 
"  OoneerrMbw  of  Boefgjr." 


380 


OBLiqtrS  IMPACT. 


any  nataral  action,  there  is  never  a  development  of  energy 
which  cannot  be  accounted  for  by  the  disappearance  of  an 
equal  amount  elsewhere  by  means  of  some  known  physical 
agency.  Hence,  the  kinetic  energy  which  appears  to  be 
lost  in  the  above  cases  of  impact,  is  only  transformed, 
partly  into  heating  the  bodies  and  the  surrounding  air,  and 
partly  into  sonorous  vibrations,  as  in  the  impact  of  a  ham- 
mer on  a  bell. 

209.  ObUqne  Impact  of  Bodiea.— The  only  other 
case  which  we  shall  treat  of  is  that  of  oblique  impact  when 
the  bodies  are  spherical  and  perfectly  smooth. 

A  particle  impinges  with  a  given  velocity,  and  in  a  given 
direction,  on  a  smooth  plane ;  required  to  determine  ike 
motion  after  impact. 

Let  AC  represent  the  direo* 
tion  of  the  velocity  before  im- 
pact, me^ng  the  plane  at  0, 
and  CB  the  direction  after 
impact.  Draw  CD  perpen- 
dicular   to  the  plane ;    then 

since  the  plane  is  smooth  its  impulsive  reaction  will  be 
along  CD. 

Let  V  and  v,  denote  the  velocities  before  and  after 
impact,  respectively ;  and  let  o  and  /3  denote  the  angles 
ACD  and  BCD. 

Resolve  v  along  the  plane  and  perpendicular  to  it.  The 
former  will  not  be  altered,  since  the  impulsive  force  acts 
perpendicular  to  the  plane ;  the  latter  may  be  treated  as  in 
the  case  of  direct  impact,  md  will  therefore,  after  impact, 
be  e  times  wL.,t  it  was  before  (Art.  207,  Cor.  7).  Hence, 
resolving  Vi 
have 


Ftg.87 


and  perpendicular  to  the  plane,  we 
Vt  sin  j3  =  V  sin  a,  (1) 


V|Cosj9=:  — evooso. 


(») 


[)ment  of  energy 
ppearance  of  an 
known  physical 
\\  appears  to  be 
ily  transformed, 
Duuding  air,  and 
npact  of  a  bam- 

The  only  other 
ue  impact  when 
th. 

and  in  a  given 
to  determine  the 


Ftg.a7 


reaction  will  be 

efore  and  after 
inote  the  angles 

Blar  to  it.  The 
iilsire  force  acts 
be  treated  as  in 
re,  after  impact, 
3or.  7).  Hence, 
I  the  plane,  we 

(1) 

m 


OBLIQVM  ntPACT. 

881 

Dividing  (2)  by  (1),  we  get 

cot  /3  =  —  e  cot  «. 

(8) 

Squaring  (1)  and  {%),  and  adding,  we  get 

«,«  =  t)»  (sin*  a  4-  fl»  cos*  «). 


(4) 


Thus  (3)  determines  the  direction,  and  (4)  the  magnitude 
of  the  velocity  after  impact. 

The  angle  ACD  is  called  the  angle  of  incidence,  and  the 
angle  BCD  the  angle  of  reflexion. 

Cor.  1.— If  the  elasticity  be  perfect,  or  e  =  1,  we  have 
from  (3)  and  (4), 


and 


oot  /3  =  —  cot «,  or  i3  =  ~  « ; 

t),'  =  v",  or  r,  =  V. 


(6) 
(6) 


Hence,  in  perfectly  elastic  balls  the  angles  of  incidence 
and  reflexion  are  numerically  equal,  and  the  velocities  before 
and  after  impact  are  equal.  This  is  the  ordinary  rule  in 
the  case  of  a  billiard  ball  striking  the  cushion. 

Cor.  2.— Suppose  e  =  0;  then  from  (8),  0  —  90". 
Thus,  if  there  is  no  elasticity,  the  body  after  impact  moves 
along  the  plane  with  the  velocity  v  sin  a. 

If  o  =  0,  so  that  the  impact  is  direct,  we  have  from  (4), 
vt  =  ev;  i.  e.,  after  the  impact  the  body  rebounded  along 
its  former  course  with  e  times  its  former  velocity. 

If  a  =  0,  and  «  =  0,  then  from  (4),  «,  =  0,  and  the 
body  is  brought  to  rest  by  the  impact. 

SoH.— Of  course  the  results  of  this  article  are  applicable 
to  oases  of  impact  on  any  smooth  surface,  by  substituting 
for  the  plane  on  which  the  impact  has  been  supposed  to 


383       OBLiqUS  IMPAC'i    OF  TWO  SMOOTH  SPBSREa, 


\'m 


vm 


m 


take  place  the  plane  which  is  tangent  to  the  surface  at  the 
point  uf  impact. 

210.  Oblique  Impact  of  Two  Smooth  Spheres.— 

Two  smooth  spkfres,  moving  in  given  directions  and  with 
given  vehcitiea,  impinge;  to  determine  the  impulse  and  the 
subsequent  motion. 

Let  the  masses 
of  the  spheres  be 
M,  M' ;  their  cen- 
tres C,  C;  their 
velocities  before 
impact  V  and  -d, 
and  after  impact 
V,  and  w,'.  Let  ED  be  the  line  which  joins  their  centres  at 
the  instant  of  impact  (called  the  line  of  impact):  CA  and 
CB  the  directions  of  motion  of  the  impinging  sphere,  M, 
before  and  after  impact ;  and  O'A'  and  C'B'  those  of  the 
other  sphere;  let  «,  a,  be  the  angles,  ACD  and  A'C'D, 
which  the  original  directions  of  motion  make  with  the  line 
of  impact;  /?,  /J,  the  angles,  BCD  and  B'C'D,  which  their 
directions  make  after  the  impact 

It  is  evident  that,  since  the  spheres  are  smooth,  the 
entire  mutual  impulsive  pressure  takes  place  in  the  line 
joining  the  centres  at  the  instant  of  impact.  Let  R  be  the 
impulse,  and  e  the  coefficient  of  restitution.  Resolve  all 
the  velocities  along  the  line  of  impact  and  at  right  angles 
to  it ;  the  latter  will  not  be  affected  by  the  impact,  and  the 
former  will  be  affected  exactly  in  the  same  way  as  if  the 
impact  had  been  direct.  Hence,  since  the  velocities  in  the 
line  of  impact  are  v  cos  «,  v'  cos  a',  t?,  cos  j3,  «,'  cos  /S",  we 
have,  by  substituting  in  (3)  and  (4)  of  Art  207, 

p,  cos  /3  =  i;  cos  «  -  jjqT^  (1  +  e)  (»  cos  a— if  cos  «'),  (1) 


r  8PHSRES. 

he  surface  at  the 


>oth  Spheres.— 

•ections  and  with 
impulse  and  the 


i  their  centres  at 
mpact):  GA  and 
iging  sphere,  M, 
I'B'  those  of  the 
CD  and  A'C'D, 
ike  with  the  line 
D'D,  which  their 

are  smooth,  the 
ilace  in  the  line 
t.  Let  R  be  the 
ion.  Resolve  all 
at  right  angles 
impact,  and  the 
ne  way  as  if  the 
Telocities  in  the 
I  j3,  «,'  cos  P",  we 
207, 

i—i/  COB  a'),  (1) 


SXAMPLSa. 

M 


383 


v,'co8/}'  =  v'oosa'+^-^y,  (l-i-e)  («oo8«-r'co8  «'),  (a) 

which  are  the  final  velocities  c/the  two  spheres  along  the  line 
of  impact  ED. 
Also,  from  (6)  of  Art.  306,  we  obtain  by  substitution, 

~  W+l^'  '"  °°''  **  —  *'  ''OS  a'),  (8) 

(See  Tait  and  Steele's  Dynamics  of  a  Particle,  p.  323.) 

Cob.  1.— Multiplyin/j  (1)  by  M,  and  (8)  by  M',  and  add- 
ing we  get 

Mv,  cos  ^  +  M'vt'  cos  i3'  =  Mv  cos  a  +  M'v^'  cos  «',  (4) 

which  shows  that  the  momentum  of  the  systtm  resolved 
along  the  Urn  of  impact  is  the  same  after  impact  as  before. 

Cor.  2.— Subtracting  (2)  from  (1)  we  obtain, 

t>,  COS  ^  —  r/  cos  /3'  =  —  e  (»  cos  a  —  v'  cos  «').     (6) 

That  is,  the  relative  velocity,  resolved  along  the  line  of 
impact,  after  impact  is  —  e  times  its  value  before. 

EXAMPLES. 

1.  A  body*  weighting  3-  lbs.  moving  with  a  velocity  of 
10  ft.  per  second,  impinges  on  a  body  weighing  2  lbs.,  and 
moving  with  a  velocity  of  8  ft  per  second ;  find  the  com- 
mon velocity  after  impact  Ans.  7\  ft  per  second. 

2.  A  body  weighing  7  lbs.  moving  ll  ft  per  second, 
impinges  on  another  at  rest  weighing  15  lbs.;  find  the  com- 
mon  velocity  after  impact  Ans.  3^  ft  per  second. 

•  The  liodiM  are  inelaaUc  aalen  otherwiM  itatcd.  The  flnt  W  ezamplee  are  in 
Mnet  Impact. 


384 


EXAMPLES, 


hiu}  \ 


3.  A  body  weighing  4  lbs.  moving  9  ft  per  second, 
impinges  on  another  body  weighing  2  lbs-  and  moving  in 
the  opposite  direction  with  a  velocity  of  5  .ft.  per  second ; 
find  the  common  velocity  after  impact. 

Ana.  4^  ft.  per  second. 

4.  A  body,  M',  weighing  6  lbs.  moving  7  ft  per  second, 
is  impinged  upon  by  a  body,  M,  weighing  6  lbs.  and  mov- 
ing  in  the  same  direction;  after  impact  the  velocity  of  M' 
is  doubled :  fi ad  the  velocity  of  M  before  impact 

An$.  19f  ft.  per  second. 

6.  Two  bodiea,  weighing  2  ll».,  and  4  lbs.,  and  moving  in 
the  same  direction  with  the  velocities  of  6  and  9  ft.  respec- 
tively, impinge  upon  each  other;  find  their  common 
velocity  after  impact  Ans.  S  ft  per  second. 

6.  A  weight  of  2  lbs.,  moving  with  a  velocity  of  20  ft 
per  second,  overtakes  one  of  6  lbs.,  moving  with  a  velocity 
of  5  ft  per  second ;  find  the  common  velocity  after  impact. 

Ana.  94  ft  per  second. 

7.  If  the  same  bodies  met  with  the  same  velocities  find 
the  common  velocity  after  impact 

Ans.  24  ft  per  second  iu  the  direction  of  the  first 

8.  Two  bodies  of  different  mtiSRcs,  are  moving  towards 
each  other,  itnth  velocities  of  10  ft  and  12  ft.  per  second 
respectively,  and  continue  to  move  after  impact  with  a 
velocity  of  !•  2  ft  per  second  in  the  direction  of  the  greater; 
compare  their  masses.  Arts.  As  3  to  2. 

9.  A  body  impinges  on  another  of  twice  its  mass  at  rest: 
■how  that  the  impinging  body  lo,  -s  two-thirds  of  its 
velocity  by  the  impact 

10.  Two  bodies  of  unequal  masses  1  viag  in  opposite 
directions  with  momenta  numerically  equal  meet;  show 
that  the  momenta  are  numerically  equal  after  impact. 


Mai 


ii 


J  ft  per  second, 
3.  aud  moving  in 
5 .ft.  per  second; 

ft  per  second. 

7  ft.  per  second, 
g  6  lbs.  and  mov- 
lie  velocity  of  M' 
mpact 
ft.  per  second. 

s.,  and  moving  in 
and  9  ft.  respec- 
their   common 
ft  per  second. 

velocity  of  20  ft 
:  with  a  velocity 
nty  after  impact 
ft  per  second. 

te  velocities  find 

on  of  the  first. 

moving  towards 
2  ft.  per  second 
'  impact  with  a 
n  of  the  greater; 
\n3.  As  3  to  2. 

its  mass  at  rest: 
ro-tbirds   of   its 


ring  in  opposite 
lal  meet;  show 
her  impaot 


EXAMPLS& 

11.  A  bodr,  M,  weighing  10  lbs.  moving  8  £1.  per  second, 
impivjges  on  M',  weighing  6  lbs.  and  moving  in  the  same 
direction  5  ft.  per  se^wnd ;  find  tbeir  velocities  after  impact, 
supposing  6  =  1. 

Am.  Velocity  of  if  =  6| ;  velocity  otM'  =  8f. 

12.  A  body,  M,  weighing  4  lbs.  moving  6  ft  per  neoond, 
meets  M'  weighing  8  lbs.  and  moving  4  ft  per  second; 
find  their  velocities  after  impact,  «  =  1. 

Ans.  Each  body  is  reflected  back,  M  with  a  velocity  of 
7|  andif '  with  a  velocity  of  2|. 

13.  Two  balls,  of  4  and  6  lbs.  weight,  impinge  on  each 
other  when  moving  in  the  same  direction  with  velocities  of 
9  and  10  ft  respectively  ;  find  their  velocities  after  impact, 
supposing  e  =  ^.  Am.  10-08  and  9-28. 

14.  Find  the  kinetic  energy  lost  by  impact  in  example  5. 

Ans.  f^. 

IL  Two  bcJies  weighing  40  and  60  lbs.  and  moving  in 
the  same  direction  with  velocities  of  16  and  26  ft  respec- 
tively, impinge  on  each  other:  find  the  loss  of  kinetic 
energy  by  impact.  Ana.  37 -3. 

10.  An  arrow  shot  from  a  bow  starts  off  with  a  velocity 
of  120  ft  per  second;  with  what  velocity  will  an  arrow 
twice  as  heavy  leave  the  bow,  if  sent  off  with  three  times 
the  force  ?  Ana.  180  ft  per  second. 

17.  Two  balls,  weighing  8  ozs.  and  6  ozs.  respectively, 
arc  feimultanoously  projected  upwards,  the  former  rises  to  a 
height  of  324  ft.  and  the  latter  to  266  ft ;  compare  the 
forces  of  projection.  Ana.  As  3  to  2. 

18.  A  freight  train,  weighing  200  tons,  and  traveling  20 
miles  per  hr.  runs  into  a  passenger  train  of  60  tons,  stand- 
ing on  the  same  track;  find  the  velocity  at  which  the 
remains  of  the  passenger  train  will  be  prcpelleO  along  the 
tnwk,  snpposing  «  =  f  Aiu.  19- 3  miles  per  hr. 


386 


^^J[AMFL£8. 


19.  Thejie  is  a  row  of  ten  perfectly  elastic  bodies  whose 
masses  increase  geometrically  by  the  constant  ratio  3,  and 
the  first  impinges  on  t  second  with  the  velocity  of 
6  ft  per  second ;  find  the  velocity  of  the  last  body. 

Ana.  -yf ,  ft  per  second. 

20.  A  body  weighing  5  lbs.  moving  with  a  velocity  of  14 
ft.  per  second,  impinges  on  a  body  weighing  3  lbs.,  and 
moving  with  a  velocity  of  8  ft.  per  second;  find  the  veloci- 
ties after  impact  supposing  e  =  \.  Ana.  11  and  13. 

21.  Two  bodies  are  moving  in  the  same  direction  with 
the  velocities  7  and  6  :  and  after  impact  their  velocities 
are  6  and  6;  find  c,  and  the  ratio  of  their  masses. 

Ans.  e  =  4;  J/'  =  2M. 

22.  A  body  weighing  two  lbs.  impinges  on  a  body  weigh- 
ing one  lb.;  e  is  ^,  show  that  v,  =  r  ^-  v',  and  that  v,'  =  v. 

23.  Two  bodies  moving  with  numerically  equal  velocities 
in  opposite  dirv'KJtious,  impinge  on  each  oth  r;  the«result  is 
that  one  of  them  turns  hack  with  its  original  velciity,  and 
the  othgr  follows  it  with  half  that  velocity;  show  that  one 
body  is  four  times  as  heavy  as  the  other,  and  tluit  e  =  J. 

24.  A  strikes  B,  which  is  at  reist,  and  after  impact  the 

velocities  are  Dumerically  equal ;  if  r  be  the  ratio  of  B'g 

2 

ma9s  to  A'b  mass,  show  that  e  is  7,  and  that  B's  mass 

r  —  1 

is  at  least  three  times  A's  mass. 

25.  A  body  impinges  on  an  equal  body  at  rest ;  show 
that  the  kinetic  energy  before  impact  caunot  be  greater 
than  twice  the  kinetic  energy  after  im^Mct 

26.  A  beries  of  perfectly  elastic  balls  are  arranged  in  the 
same  straight  line ;  one  of  them  impinges  on  the  next, 
then  this  on  the  noxt  and  so  on ;  show  that  if  their  masses 
form  a  geometric  progression  of  which  the  oommou  ratio 


atic  bodies  whose 
tant  ratio  3,  and 
the  velocity  of 
wt  body, 
ft  per  second. 

1  a  velocity  of  14 
;hiiig  3  lbs.,  and 
;  find  the  velocl- 
[ns.  11  and  13. 

le  direction  with 
!t  their  velocities 
masses. 
4;  J/'  =  2M. 

on  a  body  weigh- 
and  that  v/  =  v. 

y  equal  velocities 
n:v;  the<resnlt  ia 
inal  velciiiy,  and 
;  show  that  one 
id  tliitt  e  =  |. 

after  impact  the 
the  ratio  of  B'g 

ind  that  B's  mass 


ly  at  rest ;  show 
Eiunot  be  greater 

I  arranged  in  the 
![eH  on  the  next, 
t  if  their  masses 
he  oommou  ratio 


KXAMPLS8, 


m 


is  r,  their  velocities  after  inajiact  form  a  geometric  progres- 


sion  of  whicli  the  common  ratio  is 


r  4-  1 


27.  A  ball  falls  from  rest  at  a  height  of  20  ft.  above  ft 
fixed  horizontal  plane;  find  the  height  to  which  it  will 
rebound,  e  being  J,  and  g  being  32.  Arts.  1\\  feet. 

28.  A  ball  impinges  on  an  equal  ball  at  rest,  the  elas- 
ticity being  perfect;  if  «he  original  direction  of  the  strik- 
ing ball  is  inclined  at  an  angle  of  45"  to  the  straight  line 
joining  the  centres,  determine  the  augle  between  the 
directions  of  motion  of  the  striking  ball  before  and  after 
impmt.  Ans.  45°. 

29.  A  ball  falls  from  a  height  h  on  a  horizontal  plane, 
and  then  rebounds;  find  the  height  to  v/hicb  it  rises  in  its 
iiscent.  Ana.  e*//. 

30.  A  ball  of  mass  M,  impinges  on  a  ball  of  mass  M',  at 
i"est ;  show  that,  the  tangent  of  tlie  augle  between  the  old 
and  new  directions  of  the  motion  of  the  impinging  body  is 

1  4-  0 iTjin^o 

%        ~M  +  M'  (sin*  a  —  0  cos*  a) 

31.  A  ball  of  mass  M  impinges  on  a  ball  of  mass  M'  at 
rest ;  find  the  condition  in  order  that  the  directions  of 
motion  '^f  the  impinging  ball  before  and  after  impact  may 


be  at  right  angles. 


Ana.  tan"  n  = 


Me 

M'  H  Af' 


83.  A  ball  impinges  on  an  equal  ball  at  rest,  the  angle 
between  the  old  and  new  directions  of  motion  of  the 
impinging  ball  ia  C0° ;  find  the  velocity  after  impact,  » 
being  1.  Ang.  v  sin  30°. 

33.  A  ball  impinges  on  an  equal  ball  at  rebt,  e  being  1 ; 
liu'i  the  conditiou  under  which  the  velocities  will  ht;  vi\uaX 
uftev  impact.  Ant.  u  ^^  45° 


n 


388 


MXAMPLM8. 


34.  A  ball  is  v.<ojected  from  the  middle  point  of  one  Bide 
of  a  biUiard  table,  eo  as  to  strike  first  an  adjacent  side,  and 
then  the  middle  point  of  the  side  opposite  to  that  from 
M'hich  it  started ;  find  wk  iro  the  ball  must  hit  the  adjacent 
side,  its  length  being  b. 

Am.  At  the  distance  -^ — : —  from  the  eud  nearest  the 


opposite  aide. 


l  +  « 


W' 


e  point  of  one  side 
adjacent  side,  aud 
osite  to  that  from 
3t  hit  the  adjacent 

)  eud  nearest  the 


m 


CHAPTER    V. 

WORK    AND    ENERGY. 

211.  Deflnitioii  and  BfA^sora  of  Work.— Work  i» 
the  production  of  motion  against  rmstanee.  A  force  is  said 
to  do  work,  if  it  moves  the  hody  to  which  it  is  af  plied ; 
and  the  work  done  by  it  is  moasared  by  the  product  of  the 
force  into  the  B{>aoe  through  which  it  moves  the  body 
(Art  101,  BciD.). 

Thus,  the  work  done  in  lifting  a  weight  throagh  a  ver- 
tical distance  is  proportional  to  the  weight  lifted  and 
t)ie  vertiukl  distance  through  which  is  is  lifted.  The  unit 
of  work  used  in  England  imd  in  this  country  «'«  thtU  which 
is  required  to  overcome  the  weight  of  a  pound  through  the 
vertical  height  of  a  foot,  and  is  called  a  foot-pound.  For 
instance,  if  a  weight  of  10  lbs.  is  raised  to  a  height  of 
5  ft.,  or  5  lbs.  raised  to  a  height  of  10  ft,  50  foot-ponnds  of 
work  must  have  been  expended  in  overcoming  the  resist- 
ance of  gravity.  Similarly,  if  it  requires  a  force  of  50  lbs. 
to  move  a  load  on  a  horizontal  plane  over  a  distance  of 
100  ft.,  5000  foot-pounds  of  work  must  have  been  done. 
If  a  carpenter  urges  forward  a  plane  through  3  ft.  with  a 
force  of  13  Ibe.,  he  does  36  foot-pounds  of  work  ;  or,  if  a 
weight  of  1  lbs.  descends  through  10  ft.,  gravity  does 
70  foot-pounds  of  work  on  it, 

Her^e,  the  number  of  units  of  work,  or  foot-pounds, 
ne  <essary  to  overcome  a  constant  resistance  of  P  pounds 
through  a  distance  of  S  feet  is  equal  to  the  product  PS. 

From  this  it  appears  that,  if  the  point  of  application 
move  always  perpendicular  to  cne  direction  in  which  the 
force  acts,  such  a  force  does  no  work.  Thus,  no  work  ui 
done  by  gravity  in  the  caf:e  of  a  particle  moving  ou  % 


390 


roRK  DOXE  itr  A  Foncs. 


liurizontal  plane,  and  when  a  particle  moves  on  any  smooth 
curface  no  work  is  done  by  tlio  force  which  the  surface 
e.ierts  upon  it. 

Neither /ore?  nor  motion  alone  is  sufficient  to  constitute 
work;  so  that  a  man  who  merely  Hupports  a  load  without 
niiiving  it,  does  no  work,  in  the  sense  in  which  that  term  is 
used  mechanically,  any  more  than  a  column  does  which 
sustains  a  heavy  weight  npon  its  summit. 

If  a  body  is  moved  in  the  direction  opposite  to  that  in 
which  its  weight  acts,  the  agent  raising  it  does  work  ufKin 
it,  while  the  work  done  by  the  earth's  attraction  is  neffa- 
tii'e.  When  the  work  done  by  a  force  is  negative,  i.  e., 
when  the  point  of  application  moves  in  the  direction  oppo- 
cite  to  that  in  which  the  force  acts,  this  is  frequently 
expressed  by  saying  that  work  is  done  against  the  force. 
In  the  above  case  work  is  done  by  the  force  ?  if  ting  the 
body,  Kud  against  the  earth's  attraction. 

212.  General  Caae  of  Work  done  by  a  Force.— 

When  either  the  magnitude  or  direction  of  a  force  varies,  or 
if  l)oth  of  tliom  vary,  the  work  done  by  the  force  during  any 
finite  displacement  cannot  be  defined  as  in  Art.  Sill.  In 
this  case  the  work  done  during  any  indefinitely  small  dis- 
placement  may  be  found  by  supposing  the  magnitude  and 
direction  of  the  force  constant  during  the  displacement,  and 
finding  the  work  done  as  in  Art.  211 ;  then  taking  the  sum 
of  all  8uoh  elements  of  work  done  during  the  consecutive 
HinuU  displacements,  which  t^>gethcr  make  up  the  finite 
(lispliicemeut,  we  obtain  the  whole  work  done  by  the  force 
during  such  finite  displacement. 

TliuB  let  a  force,  P,  tot  a(  a  p<^t,  O,  In  the  direction  OP  (Fig.  ffO), 
and  let  us  Buppose  tlie  point,  O,  to  move  into  any  other  poaitiun,  A, 
very  near  0.  If  0  Iw  tiie  angl'^  betwwn  the  direction,  OP,  of  tlie 
I'orce  and  tlie  direction,  OA,  of  the  diiplaoement  of  tlie  point  of  appli- 
cation, then  the  product,  POA  ocmH,\B  called  the  work  done  by  the 
forco.  If  we  drop  a  porp«>ndlcalar,  AN,  on  OP,  the  work  done  l>y  the 
force  i8  alio  equal  to  the  pruduet  P  ■  ON,  where  ON  ii  to  lie  eati- 


:jiia"^:iPMMiiJiifly4&Myi: 


iCS. 


MEABURE  OF  WORK. 


391 


oves  on  any  smooth 
which  the  Burfaoe 

icient  to  constitute 
orts  a  load  without 
I  which  that  term  is 
jolumn  does  which 
t. 

opposite  to  that  in 
it  does  work  u[)on 
attraction  is  nega- 
a  is  negative,  /.  c, 
tiie  direction  oppo- 
,  this  is  frequently 
)  agaim,.  the  force, 
he  force  Jifting  the 

M  by  a  Force.— 

of  a  force  varies,  or 
he  force  during  any 
8  in  Art.  ail.  In 
lefinitely  small  dis- 
the  magnitude  and 
B  displacfment,  and 
len  taking  the  sum 
ag  the  consecutive 
[lake  up  the  finite 
done  by  the  force 

direction  OP  (Fig.  SO). 
vay  other  poritiun,  A, 
direction,  OP,  of  tlie 
1  "f  tJie  point  of  appli- 
the  work  done  by  the 
the  work  done  liy  the 
lere  ON  is  to  l>e  wti- 


mated  as  positive  when  in  the  direction  of  the  lorce.  If  several  forces 
act,  the  work  done  by  each  can  be  found  in  the  same  wiiy  ;  and  the 
sam  of  all  tlieoe  is  the  work  done  by  the  whole  gyatem  of  force». 

It  appears  from  this  that  tlie  work  done  by  any  force  during  an 
infinitesimal  displacement  of  the  point  of  application,  is  the  product 
of  the  resolved  part  of  the  force  in  the  direction  of  the  displacement 
into  the  displacement ;  and  this  is  the  same  as  the  virtual  momtnt  of 
tlie  force,  which  has  been  described  in  Art  101.  In  Statics  we  are 
concerned  only  with  the  small  hjfputJutkai  displacement  which  we 
give  the  point  of  application  of  the  force  in  appIyiMg  the  principle  of 
virtual  velocities.  But  in  Kinetics  tlie  liodies  are  in  motion ;  the 
force  aetitaUff  disphues  its  point  of  application  in  such  a  manner  that 
the  displacement  has  a  projection  along  the  direction  of  the  force.  If 
ds  denote  the  projection  of  any  elementary  arc  of  a  curve  along  the 
direction  of  P,  the  work  done  by  P  in  this  displacement  is  Pd».  The 
sum  of  all  these  elements  of  work  done  by  P  in  its  motion  over  a 
finite  space  is  the  whole  work  found  by  taking  the  integral  of  Pd$ 
between  proper  limits. 

Hence  generally,  if  «  be  an  arc  of  the  path  of  a  particle,  P  the 
tangential  component  of  the  for«»c  •-aich  act  on  it,  the  work  done  on 
the  particle  between  any  two  points  of  its  path  is 

/Pd»,  (1) 

the  integral  being  taken  between  limits  corresponding  to  the  initial 
and  final  positions  of  the  particle. 

213.  Work  on  an  Inclined  Plane.— Let  a  be  the 

inclination  of  the  plane  to  the  horizon,  W  the  weight 
moved,  «  the  distance  along  the  plane  through  which  the 
weight  is  moved.  Resolve  W  into  two  components,  one 
along  the  plane  and  the  other  perpendicular  to  it;  the 
former,  W  sin  a,  is  the  component  which  resists  motion 
along  the  plane.  Hence  the  amount  of  work  required  'o 
draw  the  weight  up  the  plane  =  W  sin  « •  s  =  Wxtho 
vertical  height  of  the  plane ;  t.  e.,  the  amount  of  work 
required  is  unchanged  by  the  substitution  of  the  oblique  path 
for  the  vertical.  Hence  the  work  in  moving  a  body  up  an 
incUntnl  plane,  without  friction,  is  equal  to  the  product  of 
the  weight  of  the  body  by  the  vertical  height  through  which' 
it  ia  raised. 


4 


\'i 


393 


WORK  ON  AN  INCLINHD  PLANW. 


I 


Cob.  1. — If  the  plane  be  rough,  let  ft  =  the  coefficient 
of  friction ;  then  since  the  normal  component  of  the  weight 
is  FF  COB  a,  the  resistance  of  friction  is  /i  fF  cos  a  (Art  92). 
The  work  required  consists  of  two  parts,  (1)  raising  the 
weight  along  the  plane,  and  (2)  overcoming  the  resistance 
of  friction  along  the  plane,  the  former  =  (F  sin  «  • «,  and 
the  latter  is  ^  H^  cos  « • «.  Hence  the  whoh  work  necessary 
to  move  the  weight  up  the  plane  is 


FT  (sin  a  4-  |t*  cos  cc)  «. 


(1) 


Since  »  sin  a  represents  the  vertical  height  through 
which  the  weight  is  raised,  and  »  cos  a  the  korwmtal  space 
through  which  it  is  drawn,  tiiis  result  may  be  stated  thus : 
The  work  expended  it  the  same  as  that  which  would  be 
required  to  raise  the  weight  through  the  vertioal  height  of 
the  plane.,  together  with  that  which  would  be  required  to 
draw  the  body  aUmg  the  base  qf  the  plam  horitojUaUif 
against  friction. 

GoR.  3. — If  a  body  be  dragged  through  a  space,  s,  down 
an  inclined  plane,  which  is  too  rough  for  the  body  to  slide 
down  by  itself,  the  work  done  is 


fT  (^  cos  a  —  sin  «c)  s. 


(») 


Cor.  3. — If  h  —  the  height  of  the  inclined  plane,  and 
d  =  its  horizontal  base,  then  the  work  done  against  gravity 
to  move  the  body  up  the  plane  =  Wh ;  and  the  work  done 
against  friction  to  move  the  body  along  the  plane,  suppos* 
ing  it  to  be  horixontal,  =  nb  W.  Hence  (Cor.  1)  the  total 
work  done  is 

Wh  +  yhW.  (8) 

If  the  body  be  drawn  down  the  phne,  the  total  work 
expended  (Cor.  2)  is 

-  Wh  +  ftbW.  (4) 


~ 


:  the  coefficient 
at  of  the  weight 
C08  a  (Art.  92). 
(1)  raising  the 
[  the  resiBtance 
tF  sin  «  .  9,  and 
teork  necessary 

(1) 

leight  through 
\or%»ontai  spuoe 
e  stated  thas : 
ohi<^  would  be 
HmI  height  of 
be  required  to 
M  horitoiUaUif 


^Mce,  8,  down 
'  body  to  slide 


(2) 

led  plane,  and 
igainst  gravity 
the  work  done 
plane,  euppoB- 
r.  1)  the  total 

(3) 
le  total  work 


JSJCAMl'LBS. 

If  in  (4)  the  former  term  la  greater  than  the  latter, 
gravity  does  more  work  than  what  is  expended  on  friction, 
and  the  body  elides  down  the  plane  with  accelerated 
velocity. 

ScH.  1.— If  the  inclination  of  the  plane  is  small,  as  it  ia 
in  most  cases  which  occur  iu  pmctice,  as  in  common  roads 
and  railroads,  cos  a  may  without  any  important  error  be 
taken  as  equal  to  unity,  and  the  expression  for  the  work 
becomes  (Coi-s.  1  and  2) 


W^  ()">''  ±  «  Bin  a), 


(«) 


the  upper  or  lower  sign  being  taken  according  as  the  body 
is  dragged  up  or  down  the  plane. 

ScH.  2.— If  the  inclination  of  the  piano  is  small,  as  in 
the  case  of  railway  gradients,  the  pressure  upon  the  plane 
will  be  very  nearly  equal  to  the  weight  of  the  body  ;  and 
the  total  work  in  moving  a  body  atong  an  inclined  plane 
wiU  be  from  (3)  and  (4), 


1^1  W±  Wh, 


(«) 


where  nlW  k  the  work  due  to  friction  along  the  plane 
of  length  I,  and  Wh  is  the  work  due  to  gravity,  the  proper 
sign  being  taken  as  in  (5). 

EXAMPLES. 

1.  How  much  Work  is  done  in  lifting  150  and  200  lbs. 
through  the  heights  of  80  and  120  fl.  respectively. 

The  work  done  =  150  x  80  -f  200  x  120 
=  3G000  foot-pounds,  Ans. 

2.  A  body  weighing  500  lbs.  elides  on  a  rough  horizontal 
plane,  the  coefficient  of  friction  being  0.1 ;  how  much  work 
must  be  doue  against  friction  to  move  the  body  over 
100  ft.  ? 


P 


■i 


894 


sxAitPLsa. 


Here  the  friction  is  a  force  of  60  lbs.  acting  directly 
o|)iK>site  to  the  motion ;  hence  the  work  done  againgt  firio- 
tion  to  move  thd  body  ovef  100  ft  it 

60  X  100  =  6000  foot-pounds.  Am, 

t.  A  tniin  Weigtll  lOO  totisi  the  total  resistance  is  8  lbs. 
tier  ton ;  how  mnoh  mot\  mart  be  expended  in  l«ising  it 
to  the  top  of  aa  inclined  plane  a  mile  long,  the  inclination 
of  the  plane  behig  1  rertical  to  10  horizontal. 

Here  the  work  done  against  friction  (Sch.  2) 

=  800  X  6380  =r  4224000  foot-pounds, 
and  the  work  done  against  gravity 

=  8a4000*  X  6280  x  V»  =  16896000  foot-ponnds, 
10  that  the  whole  work  =  21120000  foot-pouncis. 

0 

4.  A  train  weighing  100  tons  moves  30  miles  an  hout 
along  a  horizontal  rood ;  the  resistances  are  8  lbs.  per  ton  ; 
find  the  quantity  of  work  expended  each  hour. 

Aha.  126720000  foot-pounds. 

6.  If  26  cubic  feet  of  water  are  pumped  every  6  minutes 
from  a  mine  l40  fathoms  deep,  required  the  amount  of 
work  expended  per  minute,  a  cubic  foot  of  water  weighing 
62}^  lbs.  Am.  262500  foot-iK>undB. 

6.  How  much  work  is  done  when  an  engine  weighing 
10  tons  moves  half  b  mile  on  a  horizontal  road,  if  the 
total  resistance  is  8  lbs.  per  ton. 

Atu.  211200  foot-pounds. 

7.  If  a  weight  of  1120  lbs.  be  lifted  up  by  20  men,  20  ft. 
high,  twice  in  a  minute,  huw  much  work  does  each  man 
do  per  hour  P  Am.  1344000  foot-ponnds. 

*  Uue  lou  being  3MB  Wtti.  uoIudh  otht-rwlM:  stated. 


1^ 


BOXSJt  POWXM. 


890 


aotihg  directly 
oe  againtt  firio- 

poonds,  Am. 

Istanoe  is  8  11^ 
3d  in  inising  it 
the  inclination 
il. 

2) 

rands, 


bot-poanda, 
uncis. 

miles  an  houf 
8  lbs.  per  ton ; 

ur. 

I  foot-pounds. 

ivery  5  minutes 
the  amount  of 
water  weighing 
foot-|H>undB» 

ngine  weighing 
ai  road,  if  the 

foot-pounds. 

30  men,  20  ft. 
does  each  man 
foot-pounds. 

•ted. 


8.  A  body  falls  down  the  whole  length  of  an  inclined 
plane  on  which  tiie  coefficient  of  friction  is  0.3.  The 
height  of  the  plane  is  10  ft  and  the  base  30  ft.  On  reach- 
ing the  bottom  it  rolls  horisonti^y  on  a  pluie,  having  the 
same  coefficieDt  of  friction.    Find  how  far  it  will  roll. 

Ans.  20  ft. 

9.  How  much  work  will  be  requiisd  to  pump  8000  cubic 
feet  of  water  from  a  mine  whose  depth  is  600  fathoms. 

Atu.  1600000000  horse-power. 

10.  A  hv.se  draws  150  lbs.  out  of  a  well,  by  means  of  a 
rope  going  over  a  fixed  pulley,  moving  at  the  rate  of 
2^  miles  an  hour;  how  many  units  of  work  does  this  horse 
perform  a  minute,  neglecting  friction. 

Ana.  83000  units  of  work. 

214.  Hone  Power. — It  would  be  inconvenient  to 
express  the  power  of  an  engine  in  foot-pounds,  since  Lhis 
unit  is  so  small ;  the  term  Horse  Potoer  is  therefore  used 
in  measuring  the  performance  of  steam  v<)ngines.  From 
experiments  made  by  Boulton  and  Watt  it  was  estimated 
that  a  horse  could  raise  33000  lbs.  vertically  through  one 
foot  in  one  minute.  This  estimate  is  probably  too  high  on 
the  average,  but  it  is  still  retained.  Whether  it  is  greater 
or  lets  than  the  power  of  s  horse  it  matters  little,  while  it 
is  a  power  so  well  defined.  A  Morse  Power  there/ore  means 
a  power  which  can  perform  SSOOO  foot-pounds  of  work  in  a 
minute.  Thus,  when  we  say  that  the  actual  horse  power 
of  an  engine  is  ten,  we  mean  that  the  engine  u  able  to  per- 
form 330000  foct-poantl^  of  work  per  minute. 

It  has  been  eatimated  that  |  of  the  88000  foot-pounds  would  be 
about  the  work  of  a  hone  of  avenge  Btreng^h.  A  mule  will  perform 
f  the  work  of  a  horse.  An  aaa  will  perform  about  \  the  work  of  a 
horse.  A  man  will  do  about  ^  the  work  of  a  horse,  or  about  8800 
units  of  work  per  mluiute.  See  Even'  Applied  Mech's;  also  Byrne's 
Practical  Meoh's, 


390       WORK  OF  RA1SISO  A  SVSTSM  OF  W BIO  UTS. 


21S.  Work  of  Rntoing  a  System  of  Weights.— 

Let  P,  Q,  R,  bfc  any  three  weights  at  the  distances,  p,  q, 
r,  rospeotivoly  above  a  fixed  horizoutal  plane.  Then  [Art. 
50  (3)]  or  (Art  73,  Oor.  3),  the  distances  of  the  ceutie  of 
gravity  of  P,  Q,  B,  above  this  fixed  horizoutal  plane  is 


Pp  +  Qq  +  Jir 
P+Q-{  Ji   ' 


(1) 


Now  suppose  that  the  weights  are  raised  vertically 
tlirongh  the  heights  o,  b,  c,  respectively.  Then  the  dis- 
tance of  the  centre  of  gravity  of  the  three  weights,  in  the 
new  position,  above  the  same  fixed  hcrizontal  plane  is 


P(p  +  a)  +  Q{q  +  b)  +  R{r  +  c) 
P+Q+Ji 

Subtract!  ug  (1)  from  (2),  we  have 

Pa  +  Qb  +  lie 
P+  Q  +  Ji   ' 


(2) 


(3) 


for  the  vertical  distance  between  the  two  positions  of  the 
centre  of  gravity  of  the  three  bodies. 

Now  the  work  of  raising  vertically  a  weight  equal  to  the 
sum  of  P,  Q,  Ji,  through  the  space  deuoted  by  (3)  is  the 
product  of  the  sum  of  the  weights  into  the  space,  which  is 


Pa+  Qb  +  Re, 


(4) 


but  (4)  is  the  work  of  raising  the  three  weights  P,  Q,  R, 
through  the  heights  a,  b,  c,  respectively.  In  the  same  way 
this  may  be  shown  for  any  number  of  weights. 

Herux  when  several  toeighta  are  raised  vertically  through 
different  heights,  the  whole  work  done  is  the  same  as  that  of 
raising  a  weight  equal  to  the  sum  of  the  weights  vertically 
from  the  first  position  of  their  centre  of  gravity  to  the  last 
position.     (See  Todhuutcr's  Mech'a,  p.  338.) 


Weights.— 

istunces,  p,  q, 
.  Then  [Art. 
'  the  ceiiti'o  of 
\\  plaue  is 

(1) 

led  vertically 
rhcn  the  dis- 
eigbts,  in  the 
plane  is 


£). 


(2) 


(3) 

jitioDs  of  the 

equal  to  the 
by  (3)  is  the 
yfx,  which  is 

(4) 

hts  P,  Q,  R, 
the  same  way 

cally  through 
me  as  that  of 
hts  vertically 
■y  to  the  last 


'  ■iMmmmk^-mmmm^Mo^KmMm^ 


SXAMPLSa. 


S97 


EXAM  PLES. 


1.  How  many  horse-power  would  it  take  to  raise  3  c«fc 
of  coal  a  minute  from  a  pit  whose  deptii  is  110  fathoms? 

Dopt!i  =  110  X  6  =  C60  feet. 

3  cwt.  =  112  X  3  =  336  lbs. 

Hence  the  work  to  be  done  in  a  minute 

=  600  X  336  =  221760  foot-pounds. 

Therefore  the  horse-power 

=  221760  -^  33000  =  6.72,  Ans. 

2.  Find  how  many  cable  feet  of  water  an  engine  of 
40  horse-power'  will  raise  in  an  hour  from  a  mine  80 
fathoms  deep,  supposing  a  cubic  foot  of  water  to  weigh 
1000  ozs. 

Work  of  the  engine  per  hour  =  40  x  33000  x  60  foot- 
pounds. 

Work  expended  in  raising  one  cubic  foot  of  water 
through  80  fathoms  =  i^f^  x  80  x  6  =  30000  foot- 
pounds. 

Hence  the  number  of  cubic  feet  raised  in  an  hour 

=  40  X  33000  X  60  -!-  30000  =  2640,  Ana, 

3.  Find  the  liorse-power  of  an  engine  which  is  to  move 
at  the  rate  of  20  miles  an  hour  up  an  incline  which  rises 
1  foot  in  100,  the  weight  of  the  engine  and  load  being 
60  tons,  and  the  resistance  from  friction  12  lbs.  per  ton. 

The  horizontal  space  passed  over  in  a  minute  =  1760  ft. ; 
the  vertical  space  is  one-hundredth  of  this  =  17.60  ft. 
Hence  from  (6)  of  Art.  213,  we  have 

12  X 1760  X  60-1-60  x  2240  x  17.6=1760  x  2064  foot-pounds. 


Therefore  the  horse-power 

=  1760  X  2064  +  83000  =  110.08,  Am. 

4  A  well  ig  to  be  dng  20  ft.  deep,  and  4  ft.  in  dmmetor ; 
find  the  work  in  raising  the  material,  snppw'.ng  that  a 
cubic  foot  of  it  weighs  140  lbs.  » «^       e> 

Here  the  weight  of  the  material  to  bo  raised 

=  47r  X  20  X  140  =  140  X  80rr  lbs. 

The  work  done  is  eqaiyalcKt  to  raising  this  throngh  the 
height  of  10  ft  (Art  316).    Hanoe  tU  whole  work 

=  140  X  80rr  X  10  =  lUJOOOrr  foot-pounds,  Ane. 

5.  Find  the  horse-power  of  an. engine  QMt  would  nuso 
7  tons  of  coal  per  hour  from  a  pit  whose  depth  is  a 
fathoms. 

Work  per  minute  =  ZLiH^J^liL?  .==  ^ZiaT; 

.-.    the  horse-power  =  ^^,  Atu. 

6  Required  the  work  in  raising  water  ftom  thi«o  different 
levels  whose  depths  are  a,  *.  c  fathoms  respoctirely ;  from 
the  first  A,  from  the  second  £,  'rom  the  third  C,  cubic 
feet  of  water  are  to  be  raised  per  minute. 

Work  in  raiting  water  from  the  first  kfvel 

=  62.6  J  X  c  X  6  =  376  A-a; 

and  80  on  for  the  work  in  the  other  lev»li ; 

. ••    work  per  min.  =r  ^76  (^  .a-|.  5.*+  C-c)  foot-pounds. 

7.  Find  the  horso-powor  of  an  engine  which  draws  a 
load  of  r  tons  along  a  level  road  at  the  rate  of  m  miles 


>,  Ant. 

ft  in  dicmetor ; 
ppocing  that  a 

Bd 

lis  throQgh  the 
b  work 

mds,  Ane. 

\9t  would  nieo 
om  depth  is  a 


8. 

three  different 
5ctiTely;  from 
bird  C,  cubic 


foot-pounds. 

lioh  draws  a 
e  of  M  miltHi 


fa[A.WPM,lbi, 


an  how,  lOie  Wcdon  being  p  pooadi  per  to»,  all  other 
resistances  being  ne^ciotdL 
Work  of  tbe  wgine  per  minute 

.  .     U.-F.    ~    gj^QQQ   -   3000  »  ^"« 

8.  Bequired  the  nnmber  of  horse-power  to  nuse  2260 
onWc  ft.  of  water  an  hour,  feenii  a  mine  whose  deptii  a  63 
fathoms.  J«*.  26i, 

9.  What  weight  of  coal  will  an  engine  of  4  horse-power 
raise  in  one  hour  from  a  pit  wnose  depth  is  200  ft.  P 

Am.  89600  lbs. 

10.  la  what  time  will  an  engine  of  10  ho?»^ppwer  raise 
5  tons  of  material  from  the  depth  of  182  ft.? 

Ans.  4>48  minutes. 

11.  How  many  cubic  feet  of  water  will  an  engine  of  86 
horse-power  raise  in  an  hour  from  a  mine  whose  depth  is  40 
fathoms  P  ^»»*-  4762  cubic  feet 

12.  The  piston  of  a  steam  engine  Ut  U  ins.  in  (Hametw  5 
its  stroke  is  2^  ft.  longi  it  makes  40  strokes  per  minute ; 
the  mean  pressure  of  the  ateam  on  it  is  16  lbs.  pw  sqnara 
inch;  -^vhat  number  of  foot-pouada  is  done  by  the  steam 
per  minute,  and  what  is  the  hone-power  of  the  engine  ? 

Ans.  866072  foot-po«nds ;  8-08  H.-P. 

18.  A  w<vight  of  li  tons  is  to  be  nuwd  from  a  d^th  cl 
00  fathoms  in  one  minute;  datermine  the  hone-power  of 
the  eni^na  oaipthh  of  doing  the  work. 

4n$,  80A  H.-P, 


4(K) 


MODTTL'S  OF  A  MArniNE. 


14.  The  resistance  to  the  motion  of  a  certain  body  is 
440  lbs.;  how  many  foot-pounds  must  be  expended  in 
making  tins  body  move  over  30  miles  in  one  hour?  What 
must  be  the  horse-power  of  an  engine  tliat  does  tlic  siime 
number  of  f oot-^wunds  in  the  same  time  ? 

Am.  C9G9C00U  foot-pounds ;  35^  H.-P. 

15.  An  engine  draws  a  loud  of  CO  tons  at  tlie  rate  of  20 
miles  im  hour;  the  resistances  aro  at  the  rate  of  8  lbs.  ixsr 
ton;  liud  the  horse-poV' rol     i    ■^nginp.         Ans.  :J5»C. 

1 6.  How  many  cubic  feet  of  water  will  an  engine  of  250 
horse-power  raise  \m:  minute  from  a  depth  of  200  fathoms? 

Ann.  110  cubic  ft. 

17.  There  is  a  mine  with  three  shafts  which  are  respeo- 
tlvoly  300.  450,  and  500  't.  deep;  it  is  required  to  raise 
from  tlio  first  80,  f.om  the  second  60,  from  the  tiiird  40 
cubic  ft.  of  water  per  minute;  tlnd  the  horse-power  of  the 
engine.  An*.  li34{|. 

216.  Modulus'^  of  a  Machine.~The  whole  work  \)or- 
formed  by  a  nia'jhine  consists  of  two  parts,  the  useful  work 
and   the  I-'-at  work.     The  useful  work  is  that  which  tiio 

tliat  which  is 

^.r•Q  lost  work 

'if  unavoidably 


machine  is   detiiirned    to   produce,  or   «' 
employed  in  overcoming  vaefiil  rcsiritav 
is   that  M'bich   is  not  wanted,  Imt  w'  '; 


II  produced  or  it  is  tliat  which  is  sjwni  iu  '  '  <  i^  ning  waste- 
fill  resistances.  For  instance  in  drawing  a  •.  .  of  cars,  the 
useful  work  is  perf>>rmod  in  m«»ving  the  train,  but  the  lost 
work  is  that  which  is  xlone  in  overcoming  the  friction  of 
the  train,  the  resistance  of  gravity  on  up  grades,  the  resist- 
ance of  the  nir.  etc.  The  work  appliinl  to  a  machine  is 
equal  to  the  whole  work  done  by  tl.e  machine,  both  useful 
ond  lost,  therefore  the  useful  work  is  always  less  than  the 
work  Hp|>lied  to  the  machine. 


*  Domotinet  oailad  Hffidojcj.    <An.  lOB.) 


5ertmn  body  is 

0  expended  in 

honr?    What 

does  the  same 

j;  35^  H.-P. 

the  rate  of  20 

0  of  8  lbs.  jier 
Ans.  26' C. 

engine  of  250 
200  fathoms? 
10  oiibic  ft. 

ch  are  reapeo- 
lired  to  raise 

1  the  tiiird  40 
-power  of  the 
Ins.   134|J. 

[)le  work  j)or- 

'  useful  work 

lit  which  the 

Imt  which  is 

I'e  lost  work 

unavoidably 

•sing  waste' 

of  cars,  the 

1)11 1  the  lost 

iio  friction  of 

8.  the  rosist- 

t  machine  is 

,  both  nseful 

\e8H  than  the 


• 


BXAMPLBS. 

The  Modulus  of  a  machine  is  tfte  ratio  of  the  useful  ioork 
done  to  the  work  applied.  ThiiB,  if  the  work  applied  to  mi 
engine  be  40  horse-power,  and  the  engine  delivers  only  30 
horse-power,  the  modulus  is  |,  t.  ft,  one-quarter  of  the  work 
applied  to  the  machine  is  lost  by  friction,  etc. 

Lot  W  be  the  work  applied  to  the  machine,  W,  the  use- 
ful work,  and  »»  the  modulus.  Then  wo  have  from  the 
above  definition 

W, 


m 


W 


U) 


Ii  a  machine  were  perfect,  i.  e.,  if  there  t  ere  no  lost  work, 
the  modulus  would  be  unity;  but  in  every  machine,  some 
of  the  work  Ib  lost  in  overcoming  wasteful  resistances, 
80  that  the  modulus  is  always  less  than  unity ;  and  it  is  of 
course  the  .  bject  of  inventors  and  improvers  to  biing  this 
fraction  as  neer  to  unity  aa  possible. 

EXAMPLES. 

1.  An  engine,  of  N  effective  horse-power,  is  found  to 
pump  A  cubic  ft  of  water  per  min.,  from  a  mine  a  fathomi 
deep*;  find  the  modulus  of  the  pumps. 

Work  of  the  engine  per  min.  =  88000  N  H.-P. 

*rhe  ufl«ftil  work,  or  work  expended  in  pumping  water, 

=  62.6  A  X  6a  =  375  A^o; 

hence  from  (1)  we  have 


M  = 


876  Ao  __  A^      . 
8800O- ""  88  JV"  ^'"' 


2.  There  wore  A  cubic  ft  of  water  in  a  mine  whose  depth 
is  a  fethoms,  when  an  engine  of  N  horse-power  bogap  to 
work  the  pump;  the  water  continued  to  flow  iato  the  mine 
at  the  rate  of  B  cubic  ft  yot  minute  i  required  the  time 


^  BXAMPLMB. 

in  which  the  mine  wouM  be  cleared  of  wftter,  the  modolus 
of  the  pump  being  nu 

Let  a;  =  the  number  ot  minntea  to  dear  the  mine  of 
W«Am.    Then 

weight  of  waier  to  be  pumped  =  62-  6  (A  +  Bar) ; 
work  in  pnmpiiig  water  =r  376a  (A  -|-  Bar)  foot-poundo; 
effective  work  of  the  engine  =  m-  JV.33000ar ; 
.'.    33000  mNxss  S76a  (A  +  Bar)  ; 

•"•    *  =  88mJ^-A.«'  ^** 

8.  An  engine  has  a  6  foot  ojlinder;  the  shaft  make§  80 
revolutions  per  minute;  the  average  steam  preasure  ig  25 
lbs.  per  aquMe  inch  ;  reqnisred  the  iMmw-power  when  the 
area  of  the  piaton  ia  1800  aqnare  inches,  the  modnlas  of 
the  engine  being  \^. 

Work  done  in  one  minute  =  1800  x  25  x  6  x  2  x  30 
foot-pounda.  Wo  multiply  by  twice  the  length  of  the 
atroke,  becanae  the  piaton  is  driven  both  up  and  down  in 
one  revolution  of  the  ahaft. 

The  efleotive  horse-power  =  mmmniMJ  x \\ 

=  450,  Ana. 

i.  The  diameter  of  the  piaton  of  a  steam  engine  ia  60 
ins. ;  it  makea  11  strokes  per  minute ;  the  length  of  each 
atroke  ia  8  ft. ;  the  mean  presanre  per  square  in.  is  16  lbs.* 
reqnireil  the  number  of  cubic  ft.  of  water  it  will  raise  per 
hour  from  a  depth  of  60  f^honu,  tho  modulus  of  the 
engine  being  0'  65. 

The  number  of  foot-poanda  of  naefnl  work  done  in  one  hoar  and 
•pent  In  railing  water  =  ir  X  80«  X  8  X 15  X 11  y  80  X  0  •  66.  therefore,  etc. 

Am.  7763  cubic  ft. 


nil"!- 


ife 


the  modulus 
tibe  mino  of 

+  Rr); 
KJt-poundB ; 
OOOic; 


ft  makes  30 

"esBurs  is  25 

ar  when  the 

modnlas  of 

6  X  2  X  30 
igth  of  the 
uuidownm 


mgine  is  60 
igth  of  each 
.  is  16  lbs.; 
rill  raise  per 
ulns  of  the 


one  hoar  and 
therefore,  etc. 

cubic  ft. 


MXAMFLMS. 


409 


5.  An  engine  i»  required  to  pump  lOQQOOO  fgallons  of 
water  every  12  hours,  from  a  mine  132  fSktboms  deep ;  ftod 
the  horse-power  if  the  modolns  be  {^,  faid  a  gallon  of 
water  weighs  10  lbs.  An*.  363jV  H.-P. 

6.  What  must  be  the  horre-power  of  an  engine  working 
e  hours  per  day,  to  supply  n  families  with  g  gallons  of 
water  each  per  day,  supposing  the  water  to  be  raised  to  the 
mean  height  of  h  feet,  and  that  a  gallon  of  water  weighs  10 
lbs.,  the  modulus  being  m.  ^^^        ngh „  p 


Ana. 


198000  «m 


, 


7.  Water  is  to  be  raised  from  a  mine  at  two  different 
levels,  viz.,  50  and  80  &tiiomB,  from  the  former  30  cubic  ft, 
and  firMU  the  latter  16  «nbic  ft  per  minute ;  find  the  horse- 
power of  the  machinery  that  wiU  be  required,  assaming 
the  modulos  to  be  0*  6.  Am.  61  •  12  H.-P. 

8.  The  dianieter  of  the  piston  of  an  engine  is  80  ins.,  the 
meMx  pressure  of  tiie  steam  is  12  lbs.  per  square  inch,  the 
length  of  the  stzoke  is  10  ft,  the  nitmber  of  strokes  made 
per  minute  is  11 ;  how  many  oiibio  ft.  of  water  will  it  rside 
per  minute  from  a  depth  <^  250  fatbomii,  its  modulus  being 
0*6?  Ans.  42*46  cubic  i't 

9.  If  the  engine  in  the  last  example  had  raised  55  cubic 
ft  of  water  per  minute  from  a  depth  of  250  fathoms,  what 
would  have  been  its  modulus  ?  Am.  0-  7771. 

10.  How  many  strokes  per  minute  must  the  ungine  in 
Sx.  8  make  in  order  to  raise  16  onbio  ft  otf  water  psr 
minute  from  the  given  depth  ?  Am.  4. 

11.  What  must  be  the  length  of  the  stroke  of  an  engine 
whose  modulus  is  0'  66,  and  whose  other  dimensions  and 
conditions  of  working  are  the  same  as  in  Ex.  8,  if  they  both 
do  the  same  quantity  of  useful  work  ?  .^»«.  9-  23  ft 


404 


KIXETIC  AXD  POTKyriAL    ENRROT. 


217.    Kinetic    and    Potential    Energy.      Stored 

Work.— 77ie  energy  of  a  body  means  its  power  of  doing 
work  J  and  the  total  amount  of  enenjy  posscsaed  by  the  body 
is  inmsnred  by  the  total  amount  of  work  which  it  is  capable 
of  doing  in  passing  from  its  present  condition  to  some 
standard  condition. 

Every  moving  body  possessea  energy,  for  it  can  bo  mado 
to  do  work  by  parting  with  its  velocity.  The  velocity  of 
the  body  nmy  Iw  used  for  causing  it  to  ascend  r^rtically 
against  the  attraction  of  tho  earth,  i.  e.,  to  do  work  against 
the  TCsistunce  of  gravity.  A  cannon  Ijall  in  motion  can 
penetrate  a  resisting  body ;  water  flowing  against  a  water- 
wheel  will  turn  tho  wheel ;  tho  moving  air  drives  the  ship 
through  the  water.  Wherever  we  find  matter  in  motion 
■Wo  have  a  certain  amount  of  energy. 

Energy,  as  known  to  us,  belongs  to  one  or  the  other  of 
two  classcM,  to  which  tho  pames  kinetic*  energy  and 
potential  energy  are  given. 

Kinetic  energy  is  energy  that  a  body  possmet  ir  virtue  of 
its  being  in  motion.  It  is  energy  acturlly  in  use,  energy 
ihat  is  constantly  being  spent  The  en<  rgy  of  a  ballet  in 
motion,  or  of  a  fiy-wheel  revolving  rapidly.,  or  of  a  pile- 
driver  just  before  it  strikes  the  pile,  are  examples  of  kinetic 
energy.  The  work  done  by  a  force  on  a  body  free  to  move, 
oxortod  through  a  given  distance,  is  always  equal  to  the 
corresiwnding  increase  of  kinetic  energy  [Art  189  (3)].  If 
ii  mass,  m,  is  moving  with  a  velocity,  v,  its  kinetic  energy 
is  iwi-*  [(3)  of  Art  189].  If  this  velocity  be  generated  by  a 
oonstjant  force,  P,  acting  through  a  space,  «,  wo  have, 
(Art  211) 

Ps  =   •«!»»,  (1) 

that  is,  tho  work  done  on  the  body  is  the  exact  equivalent 
of  the  kinetic  energy,   and   the  kinetic  energy  is  recon- 

*  OnIM  alao  aeltuU  iMrgy,  or  *im^  */  meUon. 


m 


r. 

r.      Stored 

wer  of  doing 
■  by  the  body 
it  is  capable 
■ion  to  some 


can  bo  made 
le  velocity  of 
id  r^rtically 
rork  against 
motion  cau 
nst  a  water- 
ves  the  ship 
r  in  motion 

the  other  of 
energy   and 

iV  virtue  of 
1  use,  energy 
a  ballet  in 
or  of  A  pile- 
OS  of  kinetic 
ree  to  move, 
$qaal  Co  the 
189(3)].  If 
lotio  energy 
Derated  by  a 
»,  wo  have, 

(1) 

b  equivalent 
gy  is  reoon- 


KINETIC  AND  POIfSNTIAL  SNBROT. 


406 


vertible  into  the  work;  and  the  exact  amount  of  work 
whiub  the  masH  m,  with  a  velocity  v,  can  dc  against  resist- 
ance befoie  its  motion  is  completely  destroyed  is  \m\fi. 
This  is  called  stored  work,*  and  is  the  amount  of  work  that 
any  opposing  force,  P,  will  have  to  do  on  the  body  before 
bringing  it  to  rest.  Thus,  when  a  heavy  fly-wheel  is  in 
rupid  motion,  a  considerable  portion  of  the  work  of  tlie 
engine  must  have  gone  to  produce  this  motion  ;  and  before 
the  engine  can  come  to  a  state  of  rest  all  the  work  stored 
up  in  the  fly-wheel,  as  well  as  in  the  other  parts  of  the 
machine,  must  be  destroyed.  In  this  way  a  fly-wheel  acta 
as  a  reservoir  of  work. 

If  a  body  of  mass  m,  moving  through  a  space  s,  change 
its  velocity  from  v  to  v^  tlie  work  done  on  the  body  as  it 
moves  through  that  space,  (Art  189),  is 


im(v* 


••o»). 


(«) 


If  the  body  is  not  perfectly  free,  ».  e.,  if  there  is  one  force 
urging  the  body  on,  and  another  force  resisting  the  body, 
the  kinetic  energy,  \mr^,  gives  the  excess  of  the  work  done 
by  the  former  force  over  that  done  by  the  latter  force. 
Thus,  when  the  resistance  of  friction  is  overcome,  the 
moving  forces  do  work  in  overcoming  this  resistance,  and 
all  the  work  done,  in  excess  of  that,  is  stored  in  the  moving 
mass. 

Potential  energy  is  energy  that  a  body  possesses  in  virtue 
of  its  position.  The  energy  of  a  bent  watch-spring,  which 
does  work  in  uncoiling ;  the  energy  of  a  weight  raised 
above  the  earth,  as  the  weight  of  a  clock  which  does  work 
in  falling ;  the  energy  of  compressed  air,  as  in  the  air-gun, 
or  in  an  air-brake  on  a  locomotive,  which  does  work  in 
expanding ;  the  energy  of  water  stored  in  a  mill-dam,'  and 
of  steam  in  a  boiler,  are  all  examples  of  potential  energy. 

*  C*n«d  ibo  mnemmiiabd  vol*,  See  Todlw  iter'*  Moclw.,  alio  Btorad  mmgj  mmI 
not  work.    Btowmi'd  Mechulca,  p.  178. 


«D6 


SXAMPLBS. 


« 


Such  energy  may  or  may  aot  be  called  ioto  ection,  it  may 
be  dcHrmant  for  years ;  Uie  power  exiflts,  but  tbo  aotion  will 
begin  only  when  the  weight,  or  the  water,  or  the  steam  is 
released.  Henoe  the  word  potential,  is  aignifipant,  as 
expressing  that  the  energy  is  in  existence,  and  that  a  now 
power  has  been  conferred  upon  it  by  the  act  of  raisin;):  or 
oonfiDing  it 

For  example  suppose  a  weight  of  1  lb.  be  projected 
Yfflrtioally  upwards  with  a  velocity  of  32 '2  ft  {ler  second. 
The  energy  imparted  to  the  body  will  carry  it  to  a  height 
of  16. 1  ft.,  when  it  will  cease  to  have  any  vwocity.  The 
whole  of  its  kinetic  energy  will  have  been  expended ;  but 
the  body  will  have  acquired  potential  energy  instead ;  i.  e., 
aie  kinetic  enwgy  of  ttie  budy  will  all  have  been  converted 
into  potential  energy,  which,  if  the  weight  be  lodged  for 
any  time,  is  stored  up  and  ready  to  be  freed  whenevc  the 
body  shall  be  permitted  to  fall,  imd  bring  it  back  to  its 
starting  point  with  the  velocity  of  32'  2  ft  per  second ;  and 
tiins  the  body  will  feaoquire  the  kinetic  eneigy  which  it 
originally  received.  Benoe  kinetic  energy  imd  potential 
energy  are  mutually  oonvoftible. 

Let  A  be  the  height  through  which  a  body  must  lUl  to 
acquire  the  velocity  v,  m  and  W  the  mass  and  weight, 
respectively.  Then  since  •»  as  2j*,  we  have,  for  the  ii<WBd 
work, 


W 


2ffA=  Wh. 


m 


Hence  we  may  say  that  the  work  stored  in  a  moving  body 
is  measured  by  the  product  of  the  weight  of  the  body  into  the 
height  through  which  it  muaifall  to  acqmre  the  vtlociiy. 

BXAMP1.B8. 

1.  Let  a  bullet  leave  the  barrel  of  a  gan  ^i^  the  velocity 
of  1000  ft.  per  second,  and  suppose  it  to  weigh  2  oas.;  find 


■k 


Dto  action,  it  may 
ut  tho  action  will 
r,  or  the  steam  is 
is  significant,  as 
,  and  that  a  now 
e  act  of  rai^inc:  or 

lb.  be  projected 

2  ft  per  second, 
ry  it  to  a  height 
my  Velocity.  The 
en  expended ;  but 
jrgy  instead ;  i.  e., 
re  been  converted 
;bt  be  lodged  {or 
•oed  whenevc  the 
ing  it  back  to  its 
L  per  second ;  and 

3  enei^gy  which  it 
rgy  and  potential 

body  most  fall  to 
mass  and  weight, 
ive,  lor  the  iU»«d 


Wh. 


(8) 


n  a  moving  body 
f  the  body  into  the 
r«  th0  velociiff. 


a  ^th  the  velocity 
weigh  2  086. ;  find 


BXAMPLS& 


the  w<»rk  stored  up  in  ^e  bullet,  uid  the  height  from  which 
it  must  fall  to  acquire  that  velocity. 
Here  we  have  from  (3)  for  the  stored  work 


2 


(1000)»  =  Wh 


2x1^ 

=  1941  foot-pounds. 
.-.    A  =  16628  feet. 

2.  A  ball  weighing  to  lbs.  is  projected  along  a  horizontal 
plane  with  the  velocity  of  «  ft.  p?>-  second  ;  what  space,  «, 
will  the  ball  move  over  before  it  comes  to  a  state  of  rest, 
the  coefBcient  of  friction  being/? 

Here  the  resistance  of  friction  is  fw,  which  acts  directly 
opposite  to  the  motion,  therefoK  the  work  done  by  friction 
while  the  body  mov%  over  8  feet  =  fws ;  the  work    t  ored 

up  in  the  ball  =  ^t^  =  ^ ;  therefore  from  (1)  we  have 


/W8  = 


8  = 


3.  A  railway  train,  weighing  7*  tons,  has  a  velocity  of  v 
ft.  per  second  when  the  steam  is  turned  off ;  what  distance, 
s,  will  the  train  have  moved  on  a  level  rail,  whose  friction 
iap  lbs.  per  ton,  when  the  velocity  is  v^  ft.  per  second  ? 

Here  the  work  done  by  friction  =  pT8',  hence  from  (2) 

we  have 

_        .    2240  7*..         „ 


1120<t>»-V) 
~  9P 


8  = 


4.  A  train  of  T  tons  descends  an  incline  of  «  ft  in , 
length,  having  a  total  rise  of  A  ft.;  what  will  be  the  velocity, 
V,  ocquued  by  the  train,  the  friction  being/*  lbs.  per  ton P 


408 


KINETIC  BNBROY  OF  A  RIGID  BODY. 


Here  we  have  (Art.  213,  Sell.  2),  the  work  done  on  the 
train  =  the  work  of  gravity  —  the  work  of  friction 

=  Z%iQ  Th  —  pTs', 

which  is  equal  to  the  work  stored  up  in  the  train.    Hence 

2240  7V» 


^ 


=  %UQTh—pT8\ 


•  ••    V  =  y/'igh  —  rh^P'- 

5.  If  the  velocity  of  the  tnun  in  the  last  example  be 
t\  ft.  per  second  when  the  steam  is  turned  off,  what  will  be 
its  velocity,  v,  when  it  reaches  the  bottom  of  the  incline  ? 

Ans.  V  =  Vn'o^  +  ^h  —  rhvgP'' 

6.  A  body  weighing  40  lbs.  is  projected  along  a  rough 
horizontal  plane  with  a  velocity  of  150  ft.  per  sec. ;  the 
cneflicient  of  friction  is  |;  find  the  work  done  against 
friction  in  five  seconds.  Ans.  3500  foot-pounds. 

7.  Find  the  work  accumulated  in  a  body  which  weighs 
300  lbs.  and  has  a  velocity  of  64  ft.  per  second. 

Ans.  19200  foot-pounds. 

2ia  Kinetie  Bnergy  of  a  Rigid  Body  revolving 
round  an  Axis. — L«t  m  be  the  mass  of  any  iMtrticle  of 
the  body  at  the  distance  r  firom  the  axis,  and  lev  u  bo  the 
angular  velocity,  which  wil;  be  the  same  for  every  particle, 
since  the  body  is  rigid;  then  the  kinetic  energy  of  m  = 
\m  (rw)«.  The  kinetic  energy  of  the  whole  body  will  bo 
found  by  taking  the  sum  of  these  expressions  for  every 
particle  of  the  body.    Hence  it  may  be  written 


£  Jf?ir«w»  =  •a'S,  mi*. 
« 


(1) 


•'-^"^w^iffimiwwiii^^ 


M 


)  BODY. 

vork  done  on  the 
of  friction 


be  train.    Hence 


;  last  example  be 
i  off,  what  will  be 
»  of  the  incline  ? 

ed  along  a  rough 
I  ft.  per  sec. ;  the 
ork  done  against 
>00  foot-pounds. 

>ody  which  weighs 

jcond. 

too  foot-pounds. 

Body  revolving 

of  any  i)article  of 
and  leu  u  bo  the 
for  every  particle, 
c  energy  of  f?»  = 
hole  body  will  bo 
ressious  for  every 
ritteu 

(1) 


EX  A  MP  LBS. 


409 


S  wr»  is  called  the  moment  of  inertia  of  the  body  about  the 
axis,  and  will  be  explained  in  the  next  chapter. 

Hence  the  kinetic  energy  of  any  rotating  body  =  f  I<o», 
where  I  is  (he  moment  of  itiertia  round  the  axis,  and  w  the 
angular  velocity. 

in  the  case  of  a  fly-tcheel,  it  is  sufficient  in  practice  to 
tieiit  the  whole  weight  as  distributed  uniformly  along  the 
circumference  of  the  circle  dtccribcd  by  the  mean  radius 
<»f  the  rim.  Let  r  be  this  radius ;  then  the  moment  of 
inertia  of  any  particle  of  the  wheel  =  mr»,  and  the  moment 
of  inertia  of  the  whole  wheel  =  Mr^,  where  M  is  the  total 

mass.  Hence,  substituting  in  (1)  wo  have  'i  Mi^,  which 
id  the  kinetic  energy  of  the  fly-wheel. 

EXAMPLES. 

1.  Two  equal  particles  are  made  to  revolve  on  a  vertical 
axis  at  the  distances  of  a  and  b  feet  from  it ;  required  tlie 
point  where  the  two  particles  must  be  collected  so  that  the 
work  may  not  be  altei-ed. 

Let  m  =  the  mass  of  each  particle,  h  =  the  distance  of 
tlio  required  point  from  the  axis,  and  w  =  the  angular  voloc- 
ity ;  then  we  have 

Work  stored  in  both  particles  =  \m  («w)»  +  |m  (Sw)* ; 

Work  stored  in  both  particles  collected  at  point  =  m  (I'w)*;  • 

.-.    m  (iw)»  =  \m  (aw)»  -f  \m  (iw)«; 

.-.    Jc  =  ViW+~^). 

This  point  is  called  the  centre  of  gyration.     (See  next 
chapter.) 

2.  The  weight  of  a  fly-wheel  is  w  lbs.,  the  wheel  makes 
n  revolutions  per  minute,  the  diameter  is  3r  feet,  diameter 


410 


SXAMPLSa, 


of  axle  a  inches,  and  the  ooeffioieot  of  friction  on  the  axle 
/;  how  many  rovolutioni,  «,  will  tho  wheel  make  before  it 
stops  ? 

Work  stored  in  the  wheel  =  ^  (~^f*, 

3^  V  60  / 

~  2g'90O  ' 
Work  done  by  friction  in  z  rovolations' 

and  when  the  whe<        is,  we  have 

J,   na         to  it*n*t* 

J       -19.        —   9  • 


12 


z  = 


000 


3.  Beqnired  the  nnmber  of  strokes,  x,  which  the  fly-wheel 
in  the  last  example,  will  give  to  a  fozge  hammer  whose 
weight  is  W  lbs.  and  lift  A  feet,  supposing  the  hammer  to 
make  one  lift  for  every  revolution  of  the  wheel 

Here  the  work  dae  to  raising  hammer  =  Whx.       .  ■ .  Ac. 


Ans.  X  = 


«>n*nM 


150^(12IFA  +  na/w) 


4.  The  weight  of  a  fly-wheel  is  8000  lbs.,  the  diameter 
20  feet,  diameter  of  axle  14  inches,  coefBcient  of  fHction 
0.2  ;  if  the  wheel  is  separated  from  the  engine  when  mak- 
ing 27  revolutions  per  minute,  find  how  many  revolutions 
it  will  make  before  it  sti^s  {g  taken  =  381.2). 

Ana.  16.9  revocations. 


'riction  on  the  axle 
heel  make  before  it 


f". 


[8 


irhich  the  fl^-wheel 
<rg&  hammer  whoae 
ng  the  hammer  to 
I  wheel 

iee.       .•  .3k. 

wn*nh* 
i2Wb  +  ira/w)' 

)  lbs.,  the  diameter 

tfBcient  of  fHction 

engine  when  mak- 

many  revolutions 

16.9  revocations. 


KXAMPLBS. 


411 


210.  Foro*  of  •  Blow.— In  order  to  express  the 
amount  of  foroo  between  the  face  of  a  hunmer,  for  in- 
Btancc,  and  the  head  of  a  nail,  we  mnst  consider  what 
weight  mast  be  lidd  upon  tho  head  of  the  nail  to  force  it 
into  th«  wood.  A  nul  requires  a  largo  force  to  pull  it  out, 
when  friction  done  is  retaining  it,  and  to  force  it  in  must 
of  course  require  a  still  larger  force. 

Now  the  head  of  the  hammer,  when  it  delivers  a  blow 
upon  the  head  of  the  nail,  must  be  capable  of  developing  a 
force  equal  for  a  short  time  to  the  oonttnued  pv^ssure  that 
would  be  produced  by  a  very  heavy  load.  Henee,  the  effect 
of  tho  hammer  may  be  cxplcmed  by  the  principles  of  ottergy. 
When  the  hammer  is  in  lotion  it  has  a  quantity  of  kinetic 
energy  stored  up  in  it,  und  when  it  comes  in  contact  with 
tlic  nail  this  energy  is  insttmtly  oonvertod  into  work  which 
forces  the  nail  into  the  wood. 

VXAIiPLBS. 

1,  iSappose  that  a  faammefr  weighs  1  lb.,  and  that  it  is 
ijnpciled  downwards  by  the  arm  with  considerable  force,  so 
that,  at  the  instant  the  bead  of  ^e  hamm^jr  reaches  the 
Tiail,  it  is  moving  witii  a  vel6dty  of  80  ft  pe^||||ond  ;  find 
tiie  force  which  the  hammer  exerts  on  t^e  nail  if  it  is 
driven  into  the  wood  one-tmth  of  an  inch. 

Let  P  be  the  force  which  the  hammer  exerts  on  the  nail, 
then  the  work  done  in  forcing  the.  nail  into  the  wo»>d  = 
P  X  ffir,  and  the  energy  stored  up  in  the  hammer 

=:  ^„^  =  »'  =  6.2. 
o4 

Since  the  work  done  in  forcing  the  nail  into  the  wood 
mnst  be  equal  to  all  the  work  stored  in  the  hammer,  (Art. 
ai7),  wehave 

—  =  6.2;    .'.    F=7441b* 


412 


EXAMPLES, 


Hence  the  force  which  the  hammer  exerts  on  the  head  of 
the  nail  is  at  least  744  lbs. 

2.  If  the  hammer  in  the  last  example  forces  the  nail  into 
the  wood  only  0.01  of  an  inch,  find  the  force  exerted  on 
the  nail.  Ans.  7440  lbs. 

Heace,  we  see  that,  nccoiding  as  the  wood  m  hanlor,  i.  e.,  accord- 
ing e.s  the  noil  enters  I088  at  each  stroke,  the  force  of  the  blow 
beconios  greater.  So  that  when  we  speak  of  the  "  force  of  a  blow,"' 
vnc.  must  B|)ccify  how  aeon  the  body  givinf;  the  blow  will  come  t) 
r^^t,  otherwise  the  term  is  meaningless.  Thus,  suppose  n  ball  of 
100  11/6.  weight  have  a  velocity  that  will  cause  it  to  ascend  1000  ft. ; 
ir'  t^io  UII  is  to  be  stopped  at  the  end  of  1000  ft.,  a  force  of  100  ll«. 
will  do  it ;  but  If  it  is  to  be  stopped  at  the  end  of  one  foot,  it  will 
u((h1  a  force  of  100000  lbs.  to  do  it ;  and  to  9top  it  at  the  cud  ot  one 
inch  will  require  t/  force  of  1200000  Iba,  and  h)  on. 

220.  Work  of  a  Water-FalL—When  water  or  any 
body  falls  from  a  given  height,  it  is  plain  that  the  work 
which  is  stored  up  in  it,  and  which  it  is  capable  of  doing,  is 
otjnal  to  that  winch  would  Ikj  required  to  raise  it  to  the 
height  from  which  it  has  fallen ;  i.  e.,  if  1  lb.  o1^  water 
descend  tl^|pfgh  1  foot  it  must  accumulate  as  much  work 
m  would  be  required  to  raise  it  through  1  foot.  Hence 
when  a  fall  of  wz^m  is  employed  to  drive  a  wator-whc"l,  or 
any  other  hydraulic  niichiue,  whose  modulus  is  given,  the 
work  done  upon  the  maciiino  is  equal  to  the  weight  of  the 
water  in  pounds  x  its  fall  in  feet  x  the  modulus  of  the 
machine. 

BX  AMPLBS. 

1.  The  breadth  of  a  stpoam  in  h  feet,  depth  n  feet,  mean 
Telocity  r  feci  per  minute,  and  the  height  of  the  fall  A  feet ; 
find  (1)  the  horse-power,  N,  of  the  water-wooel  whose 
modulus  is  m,  and  (2)  find  the  number  of  cubic  feet.  A, 
which  the  wheel  will  pump  per  minute  from  the  Iwttom  of 
the  fall  to  the  height  of  At  feoU 


erts  on  the  head  of 


9  forces  the  nail  into 

the  force  exerted  on 

Ans.  7440  lbs. 

iB  harder,  i.  e.,  acconi- 
,  the  force  of  the  blow 
f  the  "  force  of  a  Wow, ' 

the  blow  win  come  t^^ 
hu8,  suppose  n  ball  of 
le  it  to  ascend  1000  ft.: 

ft.,  a  force  of  100  Ite. 
!  end  of  one  fo»>t,  it  will 
top  it  at  the  end  ol  one 
con. 

VTien  water  or  any 
plain  that  the  work 
8  capable  of  doing,  is 
id  to  raise  it  to  the 
<•.,  if  1  lb.  o1^  water 
lulate  as  much  work 
ugh  1  foot.  Hence 
ive  a  water-whc"l,  or 
odulus  is  given,  tlie 
to  the  weight  of  the 
the  modulus  of  the 


depth  a  feet,  mean 
fht  of  tlie  fall  A  feet ; 

watcr-Wiioel  whose 
tcr  of  cubic  feet,  A, 

from  the  Iwttom  of 


SXAMPLSS. 


413 


62.6  Aht 
.-.    A  - 


(8) 


Weight  of  water  going  over  the  i*t;l  per  min.  =  62.6  abv. 
.'.    Work  of  wheel  per  min.  =  62.5  abvhm.  (1) 

•*'  ~33000     '  '  ' 

Work  in  pumping  water  per  min.  =  62.5  J  A, ; 

which  must  =  the  work  of  the  wheel  per  min.;   hence 
from  (1)  we  have 

62.5  abvhm ; 

abvhm 

2.  The  mean  section  of  a  stream  is  6  ft.  by  2  ft. ;  its 
mean  velocity  is  35  ft  jjer  muiate  ;  there  is  a  fall  of  13  ft. 
on  this  stream,  at  which  is  erected  n  water-wheel  whoso 
modulus  is  0.65  ;  find  the  horse-power  of  the  wheel. 

Ans.  6.6  H.-P. 

3.  In  how  miiny  hours  would  the  wheel  in  Ex.  2  grind 
8000  bushels  of  wheat,  supposing  each  horse-power  to  grind 
1  bushel  per  hour  ?  Ans.  1428i^  hours. 

4.  How  many  cubic  feet  of  water  must  be  made  to 
descend  the  full  per  minute  in  Ex.  2,  that  the  wheel  may 
grind  at  the  rate  of  28  bushels  jwr  hour  ? 

Am.  1749.5  cu.  ft, 

5.  Given  the  stream  in  Ex.  2,  what  must  be  the  height 
of  the  fall  to  grind  10  bushels  per  hour,  if  the  modi'lus  of 
the  wheel  is  0.4  ?  Ana.  87.7  feet. 

C.  Find  the  useful  horsc-jwwer  of  a  wat«r-wheel,  lujp- 
l)08ing  the  stream  to  bo  5  ft.  broad  and  2  ft.  deep,  and  tx> 
ilow  with  a  velocity  of  30  ft.  per  minute ;  the  'uight  of  tho 
full  being  14  ft..,  and  tliu  modulus  of  tho  machine  0.05, 

Ans.  5.2  nearly. 


I 


4U 


MXAMPLES. 


221.  The  IDntjr  of  an  'Ba^aam.'-Tkt  duty  of  an  engine 
is  the  number  of  units  of  toork  which  it  is  capable  cf  doing 
by  burning  a  given  quantity  of  fud. — It  has  been  found  by 
experiment  that,  wha^<)Ter  may  be  the  pressure  at  v,  hich 
the  steam  is  formed,  the  quantity  of  fuel  necessary  to 
evaporate  a  given  volume  of  water  is  always  nearly  the 
same ;  hence  it  is  most  advantageous  to  employ  steam  of  a 
high  presavre.* 

In  good  ordinary  engines  the  dnty  variea  between  SOOOGO  and 
600000  units  of  work  for  a  lb.  of  eoal.  The  extent  to  which  tbe 
eoonumy  of  fool  may  be  carried  is  well  illuBtrated  by  the  engines  eni> 
ployed  to  drain  the  mines  in  Cornwall,  England.  In  1816,  the 
average  duty  of  then  engines  was  20  million  units  of  work  for  a 
boshelf  of  coal :  in  1848,  by  reason  of  snocnsive  improvements,  tlie 
averagv  duty  had  beoome  60  millions,  eflSecting  a  saving  of  £80000 
per  aunun.  It  Is  stated  ^at  in  ths  ease  of  oa»  engine,  the  duty  waa 
raised  to  196  millions.  The  duty  of  the  engine  depends  largely  on 
tlie  oonatraotloa  of  the  boiler ;  1  lb.  of  eoal  in  the  Coniish  boHer 
evaporates  11}  lbs.  of  water,  while  la  a  diflferently-shaped  boiler  8.7 
IB  the  maximum.) 

EXAMPLES. 

1.  An  engine  burns  2  lbs.  of  coal  for  each  horse-^iower 
per  hour  ;  find  the  duty  of  the  engine  for  a  lb.  of  coal. 

Here  the  work  done  in  one  hour 

=  60  X  830C0  foot-po  ands ; 

therefore  the  duty  of  the  engine  =  30  x  33000  foot-pounds, 

=  990000  foot-iwunda. 

%  How  many  bushels  of  coal  must  be  expended  in  a 
day  of  24  hours  in  raising  160  cubic  ft.  of  wttt«r  \vir  minute 


-^^  tm  TaM  In  EeshanlM'  Macaslna,  in  the  .tmut  IMl. 

t  On«  bstbol  of  coal  s  64  or  M  tbi.,  depcndliif  upon  wbura  It  la.    a<i«d«<'e, 
p.  lao. 

t  Doarna  on  Ibe  BMkm  Bni^nc    p.  IT1,  and  PalrbBli.i.  ITnofol  Infor.iMtlon, 

V.  m 


MP 


'^«  duty  of  an  mgitie 
I  is  capable  of  doing 
[i  has  been  found  by 
e  pressure  at  \,hich 
)f  fuel  necessary  to 
3  always  nearly  the 
►  employ  steam  of  a 


B  between  3000GO  and 
je  extent  to  which  the 
kted  by  the  engines  eni' 
agland.  In  181S.  the 
)n  unita  of  work  for  a 
sive  improvement«,  tlie 
ig  ■  MiTlnflr  of  £80000 
#  engine,  the  dutjr  waa 
Ine  depends  largeljr  on 
In  the  Cornish  boiler 
eotlj-shaped  boiler  &7 


T  each  horse^iower 
)r  a  lb.  of  coal. 


ids; 

33000  foot-pounds, 
00  foot-jwunds. 

be  expended  in  • 
f  wat«r  j)or  minnte 


on  wbur*  it  In.    Goodc'e, 

tli.<.  Uiipful  tnror.iMlloo, 


WOSK  OF  A    VARtABLX  PORCS. 


m 


-U^ 


from  ft  depth  of  100  fathoms;  the  doty  of  the  engine 
being  60  millions  fof  a  bniifael  of  coal  ? 

Ana.  185  bnsbela. 

3.  A  steam  engine  is  reqaired  to  raise  70  cubic  ft.  of 
water  per  minute  from  a  depth  of  800  ft. ;  find  how  many 
tons  of  coal  will  be  required  per  day  of  24  hours,  supposing 
the  duty  of  the  engine  to  be  260000  for  a  lb.  of  ooal. 

Ana.  9  tona. 


222.  Work  of  a  Vavtebte  Potc«.--When  the  force 
which  performB  Work  through  a  pTen  epaee  varies,  the 
work  done  may  be  determined  by  maltiplying  the  given 
space  by  the  mean  of  all  the  variable  forces. 

Le*  AG  reprraent  "  e  spfioe  in  units 
of  feet  through  which  a  variable 
force  is  exerted.  Divide  AG  into 
six  equal  parts,  and  8n|^)ose  i*„  /»„ 
P,,  etc.,  to  be  the  forces  in  pounds 
applied  at  the  points  A,  B,  0,  etc., 
respectively.  At  thep  ■  points  draw  the  ordinatcs  y,,  y,,  y„ 
etc.,  to  represent  the  lorces  Trhioh  act  at  the  points  A,  B, 
C,  etc.  Then  the  work  done  from  A  to  B  will  be  equal  to 
the  space,  AB,  multiplied  by  the  mean  of  the  forces  /*, 
and  P^,  i. «.,  the  work  will  be  represented  by  the  area  of 
the  snr&oe  AabB.  In  like  manner  the  work  done  from 
D  to  0  will  be  represented  by  the  area  BicO,  and  so 
on  ;  so  that  the  work  done  through  the  whole  space,  AG, 
by  a  force  which  varies  continuously,  will  be  represented  by 
the  area  Aa^O.  This  area  may  bo  found  approximately  by 
IJio  ordinary  rule  of  MenaurtUion  for  the  area  of  a  curved 
surface  with  equidistant  ordinates,  or  more  accurately  by 


A     B    c    0     E    F    a 


n|.it 


Simpson's*  rule,  the  proof  of  which  wo  shall  now  give. 
223.  Simpson'*  Rnla.—Let  y,,  y„  y„  etc.,  be  the 


*  Althoanti  II  WM  not  luveutud  by  Hituimin.    Sv«  'PDdtaonter. 


416 


SlitPSOS'B  rCLS. 


equidistant  ordinatcs  (Fig.  89)  and  /  the  distance  between 
any  two  consecutive  ordiaates;  then  by  taking  the  sum  of 
the  trapezoids,  AabB,  BicC,  etc.,  we  have  for  the  area  of 
AagQ, 

i^(yi  +  Vt)  +  Viye  +  yi)  +  iUy.  +  y*)  +  etc. 
=  \i  (y.  +  2^8  +  2^8  +  2^4  +  ^i  +  2ye  +  yi)->  (0 

which  id  the  ordinary  formula  of  mensuration. 

Now  it  is  evident  that  when  the  curve  is  always  concave 
to  the  line  AG  (1)  will  give  too  small  a  result,  and  if  con- 
vex it  will  give  too  large  a  result 

Let  Fig.  90  represent  the  space  between  any  two  odd 
consecutive  ordinates,  as  Cc  and  Ee(Fig.  89);  divide  CE 
into  three  equal  parts,  CK  =  KL  =  LE,  ,  JlAJ 
and  erect  the  ordinates  Bufc  and  U,  dividing 
the  two  trapezoids  CcdD  and  Drf«E  into  the 
three  trapezoids  CckK,  KklL,  and  LfcE. 
The  sum  of  the  areas  of  these  three  trapezoids 


K  OL 

ng.M 


=  ICK  (Cc  +  2Kk  +  2U  +  E«) 

=  ^l  (Oc  +  2Kt  4-  2LZ  4-  Ee),  (since  ^CK  =  iCD  =  \l) 

=  \l  (Cc  +  4Do  +  Ec),  (since  2K*  +  2U  =  4Do),      (2) 

which  is  a  closer  approximation   for  the  area  of  CceE 
than  (1). 

Now  when  the  curve  is  concave  towards  CE,  (2)  is 
smaller  than  the  area  between  CE  and  the  curve  ckdle ;  if 
wo  substitute  foi  Do,  the  ordinate  Drf,  which  is  a  little 
greater  than  Do  and  which  is  given,  (2)  becomes 


\l  (Cc  +  4D(/  +  E«), 
whi'-'  is  u  still  closer  api>roximatiuu  than  (2). 


(8) 


distanco  between 
iking  tlie  sum  of 
B  for  the  area  of 

+  y*)  +  ete. 
f-2y,  +y,);  (i) 
tion. 


18  always  concave 
38ult,  and  if  oon- 

?cn  any  two  odd 
;.  89) ;  divide  CE 
[iE, 


ing 
[the 
eE. 
>ids 


K  OL       E 


d  =  4Do),      (2) 
le  area  of  CwE 

rards  CE.  (2)   is 
c  curve  ckdle ;  if 
which  is  a  little 
icomca 


(8) 


(2). 


Simil    -y  we  have  for  the  areas  of  AccC  and  "EetfO, 

V  (Aa  +  4B6  +  Cc),  and  \l  (Ee  +  4F/  +  G^). 

Adalag  (3)  and  (4)  together,  we  have  for  an  approximate 
value  of  the  whole  area, 

V  [yi  +  ^7  +  2  (f/s  +  y,)  +  Hy,+y,  +  ,/,)],   (5) 

which  is  SiinpaoH's  Formula.  Hence  Sirapsou's  rule  for 
finding  the  area  approximately  is  the  following :  Divide  the 
abidssn,  AG,  into  an  even  number  of  equal  parts,  and  erect 
ordinates  at  the  points  of  division  j  then  add  toijethcr  the 
first  and  last  ordinates,  twice  the  sum  of  all  the  other  odd 
ordinates,  and  four  times  the  sum  of  all  the  even  ordinates  ; 
multiply  the  sum  by  otie-tfiird  of  the  comuon  distance 
between  any  two  adjacent  ordinates.  (See  Todbuntcr's 
Mensuration,  also  Tate's  Geometry  and  Meusurution,  also 
Morin's  Mech's,  by  Bouuett.) 

EXAMPLES. 

1.  A  variable  force  has  acted  through  3  ft. :  the  value  of 
the  force  taken  at  seven  successive  oqnidistaut  points, 
including  the  first  and  the  last,  is  in  lbs.  189,  151.2,  126, 
108,  94.6,  84,  75.6  ;  find  tiie  whole  work  done. 

Ans.  346,4  foot-jwunds. 

2.  A  variable  force  has  acted  through  6  ft. ;  the  Ytduo  of 
the  force  taken  at  seven  successive  equidistant  pointij, 
including  the  first  and  the  lust,  is  in  lbs.  3,  8,  15,  24,  35, 
48,  63  ;  find  the  whole  work  done. 

Ans.  162  foot-j)onnds. 

8.  A  variable  force  has  a<'tod  through  9  ft.;  the  value  of 
the  force  taken  at  se^en  successive  equidistant  points, 
including  the  first  and  the  lost,  is  in  lbs.  6.082,  6.164, 
6.245,  6.403,  6.481,  6.557;  find  the  whole  work  done. 

Ans.  56.907  foot-puund«. 


':m. 


MMMnppMKMMT^cwu^  .- 


416 


aXAMPLMS. 


Should  any  of  the  ordinates  become  zero,  it  will  not  pre- 
vent the  use  of  Simpeon's  rale. 

Simpson's  rule  is  a^Uoable  t:)  other  investigations  as 
well  as  to  that  of  work  done  by  a  variable  fOTce.  For 
example,  if  we  want  the  velocity  generated  in  a  given  time 
in  a  particle  by  a  variable  force,  let  the  straight  line  AG 
represent  the  whole  time  during  which  the  fcnrce  actii,  and 
let  the  stTMght  lines  at  right  angles  to  AG  reprei>6ui  the 
force  at  the  corresponding  instants;  then  the  area  will 
represent  the  whole  space  described  in  the  given  time. 

BXAMPL.BS. 

1.  The  ram  of  a  pile-driving  engine  weighs  half  a  ton,* 
jmd  has  a  fall  of  17  ft. ;  how  many  units  of  work  are  per- 
foi-med  in  raising  this  ram  P        Ana.  19040  foot-pounds. 

a.  How  many  units  of  woA  are  required  to  raise  7  cwt 
of  coal  from  a  mine  whose  depth  is  13  fathoms  ? 

Ans.  61162  foot-pounde. 

3.  A  horse  is  used  to  lift  the  earth  fi-om  a  trench,  which 
he  does  by  moans  of  a  cord  going  over  a  pulley.  He  pulls 
up,  twice  v^ery  5  minutes,  a  uan  weighing  130  lbs.,  and  a 
barrowf ul  of  earth  weighing  860  lbs.  Each  time  the  horse 
gooa  forward  60  ft. ;  find  the  units  of  woik  done  by  the 
horse  per  hour.  Ana.  374000. 

4.  A  railway  i  ain  of  T  tons  asoends  an  inclined  plane 
which  has  a  rise  of  e  ft.  in  100  ft.,  with  a  uniform  speed  of 
m  miles  per  hour ;  find  the  horse-power  of  the  engine,  the 
friction  being/)  lbs.  per  ton. 

Ans.  ^y(P+/^-^)  H..P. 

6.  A  railway  train  of  80  tons  ascrnds  an  incline  which 
rises  one  foot  in  50  ft.,  with  the  uniform  rate  of  16  miles 


*  Ont  lull  --  tW  ewt.  -  9S40  lbs. 


wmmmmtmtm 


mi 


rot  it  will  not  pre- 

'  investigations  as 
riable  force.  For 
d  in  a  given  time 
straight  line  AG 
he  force  acts,  and 
A.G  repreucui  the 
len  the  area  will 
e  given  time. 


reighs  half  a  ton,* 
of  work  are  per- 
'40  foot-pounds. 

d  to  raise  7  cwt 

loms  ? 

52  foot-ponnde. 

1  a  trench,  which 
pulley.  He  pulls 
g  130  lbs.,  and  a 
h  time  the  horse 
oi'k  done  by  the 
Am.  374000. 

m  inclined  plane 
miform  speed  of 
'  the  engine,  the 

+  22^)  „  p 
76 "'^^ 

an  incline  which 
rate  of  16  miles 


SXAMPCES. 


419 


per  hour ;  find  the  horse-power  of  the  engine,  the  friction 
being  8  lbs.  per  ton.  Ans.  169.96  H.-P. 

6.  If  a  horse  exert  a  traction  of  t  lbs.,  what  weight,  v>, 

will  he  pull  up  or  down  a  hill  of  small  inclination  which 

has  a  rise  of  0  in  100,  the  coefficient  being/? 

.  100/ 

Ana.  w  =  -— r-; • 

100/±« 

7.  From  what  depth  will  an  engine  of  22  horse-power 


raise  13  t«ins  of  coal  in  an  hour  ? 


Am.  24.9  ft. 


8.  An  engine  is  observed  to  raise  7  tons  of  material  an 
hour  from  a  mine  whose  depth  is  85  fothoms ;  find  the 
horse-power  of  the  engine,  supposing  |  of  its  work  to  be 
lost  in  transmission.  Ans.  4.4829  H.-P. 

9.  Required  the  horse-power  of  an  engine  that  would 
supply  a  city  with  water,  working  12  hours  a  day,  the 
water  to  bo  raised  to  a  height  of  50  ft. ;  the  number  of 
inhabitants  being  130000,  and  each  person  to  use  5  gallons 
of  water  a  day,  the  gallon  weighing  8|  lbs.  nearly. 

Aug.  11.4  H.-P. 

10.  Prom  what  depth  will  an  engine  of  20  horse-power 
raise  600  cubic  feet  of  water  per  hour  P     Ans.  1056  feet 

11.  At  what  rate  per  hour  will  an  engipe  of  30  horse- 
power draw  a  train  weighing  90  tons  gross,  the  resistance 
being  8  lbs.  per  ton  ?  Ant.  16.628  miles. 

12.  What  is  the  gross  weight  of  a  train  iHliioh  an  engine 
of  25  horse-power  will  draw  at  the  rate  of  26  miles  an 
hour,  resistances  being  8  lbs.  per  ton  ? 

Ans.  46.875  tons. 

13.  A  train  whose  gross  weight  is  80  tons  travels  at  the 
rate  of  20  miles  an  hour;  if  the  resistance  is  8  Uw. 
per  ton,  what  is  the  horse-power  of  the  engine  ?  ' 

Am.  Ui\  H.-P. 


r 


i'. 


420 


X^  AMPLSa. 


14.  What  must  be  the  length  of  the  stroke  of  a  piston 
of  an  engine,  the  surfuco  of  wliich  is  1500  square  iiicliee;, 
which  makes  20  strokes  per  minute,  so  that  with  u  weuii 
pressure  of  12  lbs.  on  each  square  aua  of  the  piston,  the 
engine  may  be  of  80  horse-power  ?  Ans.  7|  ft. 

15.  The  diameter  of  the  piston  of  an  engine  is  80  ins., 
the  length  of  the  stroke  is  10  ft,  it  makes  11  strokes  per 
minute,  and  the  mean  pressure  of  the  steam  on  the  piston 
is  12  lbs.  per  square  inch  ;  what  is  the  horse-power  ? 

Ans.  201.0CH..P. 

16.  The  cylinder  of  a  steam  engine  has  an  internal 
diameter  of  3  ft,  the  length  of  the  stroke  is  6  ft.,  it  miikcs 
6  strokes  per  minute;  under  what  efFective  pressure  ^ar 
square  inch  would  it  have  to  work  in  order  that  T5  hoi'sc- 
power  may  be  done  on  the  piston?  Ans.  67' 54  lbs. 

17.  It  is  said  that  a  horse,  walking  at  the  rate  of  2^  miles 
an  hour,  can  do  1G50000  units  of  work  in  an  hour ;  what 
force  in  pounds  does  ho  continually  exert  ? 

Ans.  126  lbs. 

18.  Find  the  horse-power  of  an  engine  which  is  to  move 
at  the  rate  of  30  miles  an  hour,  the  weight  of  the  engine 
and  load  being  50  tons,  and  the  resistance  from  friction 
10  lbs.  jier  ton.  Ans.  64  H.-P. 

19.  There  were  0000  cubic  ft.  of  water  in  a  mine  whoso 
depth  is  60  fathom^,  when  an  engine  of  50  horse-power 
began  to  work  the  pump ;  the  engine  continued  to  work  5 
hours  before  tho  mine  was  cleared  of  the  water  ;  required 
the  number  of  cubic  ft  of  water  wliich  had  run  into  the 
mine  during  the  5  hours,  supposing  \  of  the  work  of  the 
engine  to  be  lost  by  transmission.      Ans.  10500  cubic  ft 

20.  Find  the  horse-jiower  of  a  steam  engine  which  will 
raise  30  cubic  it.  of  water  ^Ksr  minute  from  a  mine  440  ft 
doop.  Ann.  25  ll.-P. 


"oke  of  a  piston 

0  square  iiiclictj, 
it  with  u  lucuii 
'  the  piston,  tbo 

Ans.  7|ft. 

igiue  is  80  ins., 
11  strokes  per 

1  on  the  piston 
-power  ? 
J01.00H.-P. 

as  an  internal 
6  ft.,  ft  miikcs 
e  pressure  ^  ar 
r  that  75  hoi-se- 
8.  67. 54  lbs. 

^to  of  2^  miles 
an  hour ;  what 

ns.   125  lbs. 

ich  is  to  move 
of  the  engine 
from  friction 

|w.  64H.-P. 

a  mine  whoso 
0  horse-powor 
uod  to  work  5 
iter  ;  required 
run  into  the 
ifi  work  of  the 
00  cubic  ft. 

36  which  will 
a  mine  440  ft. 
6.  aoU.-P. 


,jjji|i,|gi!a>;iii|^pi!ij^^iiyffyy?tjMW 


SXAMPLSS. 

21.  If  a  pit  10  ft.  deep  with  an  area  of  4  square  ft.  be 
excavated  and  the  earth  thrown  up,  how  much  work  will 
have  been  done,  supposing  a  cubic  foot  of  earth  to  weigh 
90  lbs,  Ans.  18000  ft-lbs. 

22.  A  well-shaft  300  ft.  deep  and  5  ft.  in  diameter  is  full 
of  water ;  how  many  units  of  work  must  be  expended  in 
getting  this  water  out  the  well ;  (t.  «.,  irrespectively  of  any 
other  water  flowing  in)?  Ans.  165223850  ft.-lbs. 

23.  A  shaft  o  ft.  deep  is  full  of  water;  find  the  depth  of 
the  surface  of  the  water  when  one-quarter  of  the  work 
required  to  empty  the  shaft  has  been  done.         .        a  „ 

i 

24.  The  diameter  of  the  cylinder  of  an  engine  is  80  ins., 
the  piston  makes  per  minute  8  strokes  of  lOJ  ft.  under  a 
mean  pressure  of  15  lbs.  per  square  inch  ;  the  modulus  of 
the  engine  is  0'65;  how  many  cubic  ft  of  water  will  it 
raise  from  a  depth  of  112  ft.  in  one  minute? 

Ans.  485. 78  cub.  ft. 

25.  If  in  the  last  example  the  engine  raised  a  weight  of 
66433  lbs.  through  90  ft.  in  one  miuute,  what  must  bo  the 
mean  pressure  jier  square  inch  on  the  piston  ? 

Ans.  26.37  lbs. 

26.  If  the  diameter  of  the  piston  of  the  engine  in  Ex.  24 
had  been  85  ins.,  what  addition  in  horse-power  would  that 
make  to  the  useful  power  of  the  engine  ? 

^»«.  13-28  H.-P. 

27.  If  an  engine  of  60  horse-power  raise  2860  cub.  ft.  of 
water  per  hour  from  a  mine  60  fathoms  deep,  find  the 
modulus  of  the  engine.  Atu.  '65, 

28.  Find  at  what  rate  an  engine  of  30  horse-power  could 
draw  a  train  weighing  50  tons  up  an  incline  of  1  in  280, 
the  resistance  from  friction  being  7  lbs.  per  ton.  , 

Ans,  1320  ft,  per  minute. 


422 


SXAUPLSS. 


29.  A  forge  hammer  weighing  300  lbs.  makes  100  lilts  a 
minate,  the  perpendicniar  height  of  each  lift  being  2  fL; 
what  is  the  horse-power  of  the  engine  that  gives  motion  to 
20  such  hammers?  Ans.  36*30  H.-P. 

30.  An  engine  of  10  horse-power  raises  4000  lbs.  of  coal 
from  a  pit  1200  ft.  deep  in  an  hour,  and  also  gives  motion 
to  a  hammer  which  makes  50  lifts  in  a  minute,  each  lift 
having  a  perpendicular  height  of  4  ft.;  what  is  the  weight 
of  the  hammer?  Ans.  1250  lbs. 

31.  Find  the  horse-powor  of  the  engine  to  raise  T  tons  of 

coal  per  hour  from  a  pit  'vhose  depth  is  a  fathoms,  and  \jA 

the  same  time  to  give  motion  to  a  forge  hammer  weighing 

to  lbs.,  which  makes  n  lifts  per  minute  of  h  ft  each. 

224flr  +  nhw  „  _ 
Ana,  7^-^t ■  xl.-r. 

32.  Find  the  useful  work  done  by  a  fire  engine  per 
second  which  difioharges  every  second  13  lbs.  of  water  with 
a  velocity  of  60  ft.  per  second.  Am.  508  nearly. 

33.  A  mil  way  truck  weighs  m  tons ;  a  horse  draws  it 

along  horizontally,  the  resistance  being  «  lbs.  per  ton ;  in 

passing  over  a  sjiace  a  the  velocity  changes  from  «  to  f ; 

find  the  work  done  by  the  horse  in  this  space. 

2240W  ,  .        J,    , 
Ana.  —X —  («»  —  «')  +  mns. 

34.  The  weight  of  a  ram  is  600  lbs.,  and  ai  the  end  of 
the  blow  has  a  velocity  of  32J  ft.;  what  work  has  Ijcen 
done  in  raising  it  P  Am.  9G50. 

35.  Retiuired  the  work  stored  in  a  cannon  ball  whose 
weight  is  32^  lbs.,  and  velocity  1500  ft.      Am.  1126000. 

86.  A  ball,  weighing  20  lbs.,  is  projected  with  a  velocity 
of  60  ft.  a  second,  on  a  bowling-green  ;  what  space  will  the 
ball  move  over  before  it  comes  to  rest,  allowing  the  friction 
to  be  T«5  the  weight  of  the  ball?  Am.  1007.3  ft. 


':'mimimimmmmmmf>mm 


akes  1001iit«  a 
lift  being  2  ft.; 
jives  motion  to 
36.36  H.-P. 

OOO  lbs.  of  coal 
10  gives  motion 
linuto,  each  lift 
t  is  the  weight 
ns.  1250  IbH. 

raise  T  tons  of 
ithoms,  and  "ut 
nmer  weighing 
t.  each. 
■h  nhw 


H.-P. 


ire  engine  per 
.  of  water  with 
508  nearly. 

Iiorse  draws  it 
per  ton;  in 
from  u  to  V ; 

«')  +  mns, 

at  the  end  of 
'ork  has  Ixjen 
Ans.  9650. 


ball  ^.vhose 
1125000. 

ith  a  Telocity 
space  will  the 
the  friction 
1007-3  ft. 


SXAMPLKS. 


m 


87.  A  train,  weighing  198  tons,  haa  ft  velocity  of  80 
miles  an  hour  when  the  steam  is  turned  off;  how  far  will 
the  train  move  on  a  level  nul  before  coming  to  rest,  the 
friction  being  6^  Iba  per  ton  !*  Am.  12266  ft. 

38.  A  train,  weighing  60  tons,  has  a  velocity  of  40  miles 
an  hour,  when  the  steam  is  turned  off,  how  Ux  will  it 
ascend  an  incline  of  1  in  100,  taking  friction  at  8  lbs.  a  ton  ? 

Jim.  3942i  ft 

39.  A  carriage  of  1  ton  moves  on  a  level  rail  with  the 
speed  of  8  ft  a  second;  through  what  space  must  the 
carriage  move  to  have  a  velocity  of  2  ft.,  supposing  friction 
to  be  8  lbs.  a  ton?  J >w.  348  ft 

40.  If  the  carriage  in  the  last  example  moved  over  400 
feet  before  it  comes  to  a  state  of  rest,  what  is  the  resistance 
of  friction  per  ton  ?  Ans.  6.57  lbs. 

41.  A  force,  P,  acts  upon  a  body  parallel  to  the  plane; 
find  the  space,  %  moved  over  when  the  body  has  attained  a 
given  velocity,  v,  the  coefficient  of  friction  being/,  and  the 

body  weighing  «?  lbs.  a*,,    , ?£??__. 

Ana.  "  -  2g{P  -fw) 

42.  Suppose  the  body  in  the  last  example  to  bo  moved 
for  t  seconds ;  reqair«)d  (1)  the  velocity,  v,  acquired,  and 
(2)  the  work  stored. 


A.ta.  (1) 


■1 


to 


w 


t9\  (2) 


2w 


43.  A  body,  weighing  40  lbs.,  is  projected  along  a  rough 
horizouta!  plane  with  a  velocity  of  150  ft  per  second  ;  the 
coefficient  of  friction  is  \ ;  find  the  work  done  against  fric- 
tion in  5  seconds.  Ans.  3500  foot-pounds. 

44.  A  body  weighing  60  lbs.,  is  projected  along  a  rough 
horizontal  plane  with  the  velocity  of  40  yards  per  second ; 
find  the  work  expended  when  the  body  comes  to  rest. 

Ah8.  11260  ft-lbs. 


■■yf  ^«»'»,'  '«■*■< 


424 


-..   -     EXAJtPLKS. 


' 


45.  If  a  train  of  cars  weighing  100000  lbs.  is  moving  on 
ft  horizontal  truck  with  a  velocity  of  40  milos  an  hour  when 
the  Bteam  is  turned  off ;  through  what  space  will  it  move 
before  it  ig  brought  to  rest  by  friction,  the  friction  being 
8  lbs.  i)er  ton  ?  Am.  13374.8  ft. 

40.  What  amount  of  energy  is  acquired  by  a  body  weigh- 
ing 30  lbs.  that  falls  through  the  whole  length  of  a  rough 
inclined  plane,  the  height  of  which  is  30  ft.,  and  the  base 
100  ft.,  the  coefficient  of  friction  being  \  ? 

Ans.  300ft.-lb8. 

47.  If  a  train  of  cars,  weighing  7' tons,  ascend  an  in-line 

having  a  raise  of  e  ft.  in  100  ft.,  with  the  velocity  v^  ft.  per 

second  when  the  steam  is  turned  off;  through  what  space, 

X,  will  it  move  before  it  comes  to  a  state  of  rest,  the  friction 

being j»  lbs.  per  ton  ?  .  liaOi--^ 

Ann.  X  =  — TT-r-.— ^ 

g  {•l-i.\e,  +  ;)) 

48.  Suppose  the  train,  in  Ex.  4,  Art.  217,  to  be  attached 
to  a  rope,  passing  round  a  wheel  nt  the  top  of  the  incline, 
which  has  an  empty  train  of  T,  tons  attached  to  the  other 
extremity  of  the  rope:  what  velocity,  r,  will  the  train 
acquii-e  iu  descending  s  ft.  of  the  incline  ? 


Ans,  r 


Ti 


gps 
1126" 


49.  An  engine  of  35  horse-power  makes  20  revolutions 
per  minute,  the  weight  of  the  Hy-wheel  is  20  tons  and  the 
diameter  is  20  ft.;  what  is  the  accumulated  energy  in  foot- 
pounds? Ans.  307000. 

50.  If  the  fly-wheel  ui  the  last  example  lifted  a  weight  of 
4000  lbs.  through  3  ft.,  what  part  of  its  angular  velocity 
woitlu  it  lose  ?  Ans.  ^. 

51.  If  the  axis  of  tlie  above  fly-wheel  be  6  ins.  in 
diameter,  the  coefficient  of  friction  0-075,  what  fraction, 


.  IS  moving  on 
3  an  hour  when 
aco  will  it  move 
3  friction  being 
r.  13374.8  ft. 

T  a  body  weigh- 
>gth  of  a  rough 
,  and  the  base 

?.  300ft.-lbs. 

fend  an  in':!line 
)eity  j'p  ft.  per 
jh  what  space, 
)st,  the  friction 

_l]20j;o2 

{•i'ZAe+p)' 

to  be  attached 
nf  the  incline, 
to  the  other 
tvill  the  train 


9P« 
1126' 


!0  revolutions 
tons  and  the 
lergy  in  foot- 
n».  307000. 

d  a  weight  of 
pilar  velocity 
Ans.  ■^. 

be  6  ins.  in 
'hat  fraction, 


Hi 


EXAMPLES. 


approximately,  of  the  35  horse-power  is  expended  in  turn- 
ing the  lly-wheel  ?  Ans.  ^. 

52.  In  pile  driving,  38  men  raised  a  ram  12  times  in  an 
hour ;  the  weight  of  the  ram  was  12  cwt.,  and  the  height 
through  which  it  wan  raised  140  ft.;  find  the  work  done  by 
one  man  in  a  minute.  Ans.  990  ft-lbs. 

53.  A  battering-ram,  weighing  2000  lbs.,  strikes  the 
head  of  a  pile  with  a  velocity  of  20  ft  per  second ;  how  far 
will  it  drive  the  pile  if  the  constant  resistance  is  10000  lbs.? 

Ans.  1.25  ft. 

54.  A  nail  2  ins.  long  was  driven  into  a  block  by  suc- 
cessive blows  from  a  monkey  weighing  5.01  lbs.;  after  one 
blow  it  was  found  that  the  head  of  the  nail  projected  0.8 
of  an  inch  above  the  surface  of  the  block  ;  the  monkey  was 
then  raised  to  a  height  of  1.5  ft,  and  allowed  to  fall  uiwn 
the  head  of  the  nail ;  after  this  blow  the  head  of  the  nail 
was  0.46  of  an  inch  above  the  surface ;  find  the  force  which 
the  monkey  exerted  upon  the  head  of  the  nail  at  thia  blow. 

Ans.  265.66  lbs. 

55.  The  monkey  of  a  pile-driver,  weighing  500  lbs.  is 
raised  to  a  height  of  20  ft,  and  then  allowed  to  foil  upon 
the  head  of  a  pile,  which  is  driven  into  the  ground  1  inch 
by  the  blow;  find  the  force  which  the  monkey  exerted 
upon  the  head  of  the  pile.  Ans.  120000  lbs. 

56.  A  steam  hammer,  weighing  500  lbs.,  falls  through  a 

height  of  4  ft.  under  the  action  of  its  own  weight  and  a 

steam  pressure  of  1000  lbs.;   find  the  amount  of  work 

which  it  can  do  at  the  end  of  the  fall. 

Ans.  6000  ft-lbs. 

57.  The  mean  section  of  a  stream  is  8  square  ft.;  its 
mean  velocity  is  40  ft  per  minute ;  it  has  a  fall  of  17^  ft.; 
it  is  required  to  raise  water  to  a  height  of  300  ft  by  means 
of  a  water-wheel  whose  modulus  is  0.7 ;  how  many  cubic  ft 
will  it  raise  per  minute  ?  Ans.  13.07  cub.  ft. 


-t' 


426 


XXAMPLBA 


68.  To  what  height  would  the  wheel  in  the  last  example 
raise  2^  cub.  ft  of  water  per  minute  ?        Ans.  1742}  ft 

59.  The  meaa  eectioo  of  a  stretam  is  1|  ft  by  11  ft;  ita 
ntean  velocity  is  2^  miles  an  hour  ;  there  is  on  it  a  fall  of 
6  fk.  on  which  is  erected  a  wheel  whose  modulus  is  0.7 ;  this 
wheel  is  employed  to  raise  the  hammers  of  a  forge,  each  of 
which  weighs  2  tons,  and  hw  a  lift  of  l^ft.j  how  many 
lifts  of  a  hammer  will  the  wheel  ,^eld  per  minute  P 

Am.  142  nearly. 

60.  In  the  last  example  determine  the  mean  depth  of 
the  stream  if  the  wheel  yields  136  lifts  per  minute. 

Ana.  1.43  ft 

61.  In  Ex.  59,  how  many  cubic  ft  of  water  must  descend 
the  fall  per  minute  to  yield  97  lifts  '>f  the  hammer  per 
minute  ?  Ans.  2483  cub.  ft 

62.  A  stream  is  a  ft  broad  and  b  ft  deep,  and  flows  at 
the  rate  of  e  ft  per  hour ;  there  is  a  fall  of  A  ft ;  t  lie  water 
turns  a  machine  of  which  the  modulus  is  e  ;  And  the  num- 
ber of  bushels  of  corn  which  the  machine  can  grind  in  an 
hour,  supposing  that  it  requires  m  units  of  work  per 
minute  for  one  hour  to  grind  a  bushel.      ^   _   lOOOabche 


Ans. 


16  X  60m 


63.  Down  a  l4-ft  fall  200  cnK  ft.  of  water  descend  every 
minute,  and  turn  a  wheel  whose  modulus  is  0.6.  The 
wheel  lifts  water  from  the  bottom  of  the  fall  to  a  height  of 
64  ft;  (1)  how  many  cubic  ft  will  be  thus  raised  jkt 
minute?  (2)  "  the  water  were  raised  from  the  top  of  the 
fall  to  the  same  [loint,  what  would  the  numl)cr  of  cubic  ft 
then  be?  Ans.  (1)  31.1  cub.  ft.;  (2)  34.7  cub.  ft. 

In  tlut  ispcond  caim  the  numbnr  of  cub.  ft.  of  water  taken  from  the 
top  of  tho  fnll  bi-lDg  <r,  tbe  number  of  ft.  tliAt  will  turn  the  wheel  will 
bcaOO-sr. 


04.  Find  how  many  units  of  work  are  stored  up  in  a 


.•M 


last  example 
19.  174iJfft. 

by  11  ft. ;  ita 
>n  it  a  tall  of 
las  is  0.7 ;  this 
forge,  each  of 
!t.j  how  many 
lute? 

142  nearly. 

leao  depth  of 

nute. 

fu.  1.43  ft. 

'  mast  descend 
I  hammer  per 
1483  cub.  ft. 

and  fiowH  at 
ft;  die  water 
Ind  the  num- 

grind  in  an 
of  work  per 

lOOOabcJte 

16  X  60m' 

descend  every 
is  0.6.  The 
to  a  height  of 
us  raised  jwr 
ic  top  of  the 
r  of  cubic  fU 
4.:  cub.  ft. 

takeu  from  the 
the  wheel  will 


Drcd  up  iu  a 


sm 


IHiiiJiMIIUL, 


KXAMPhSa. 


427 


mill-pond  which  is  100  ft.  long,  60  ft.  br^,  «nd  3  ft.  deep, 
and  has  a  fall  of  8  ft.  Ans.  7600000. 

66.  There  are  three  distinct  levels  to  be  pamped  in  a 
mine,  the  flrst  100  fathoms  deep,  the  second  120,  the  thiid 
150 ;  30  cub.  ft.  of  w&ter  are  to  come  from  the  first,  40  from 
the  second,  and  60  from  the  third  per  minute ;  the  duty  of 
the  engine  is  70000000  for  a  bushel  of  coal  Determine  (1) 
its  working  horse-power  and  (2)  the  consumption  of  coal 
per  hour.  Ana.  (1)  191  H.-P. ;  (2)  5.4  bushels. 

66.  In  the  last  example  suppose  there  is  another  level  of 
160  fathoms  to  be  pumped,  that  the  engine  docs  as  much 
work  as  before  for  the  other  levels,  and  that  the  utmost 
r»ower  of  the  engine  is  276  H.-P. ;  find  the  greatest  number 
of  cub.  ft.  of  water  that  can  be  raised  from  the  fourth  level. 

An$.  46^  cub  ft. 

67.  A  variable  force  has  acted  throngh  8  ft.;  the  value 
of  iho  force  taken  at  nine  successive  equidistant  points, 
including  the  first  and  the  last,  is  in  lbs.  10.204,  9.804, 
9,434,  9.090,  8.771,  8.475,  8.197,  7.937,  7.692;  find  the 
whole  work  done.  Ana.  70.641  foot-pounds. 

68.  The  value  of  a  variable  force,  taken  at  nine  succes- 
sive equidistant  points,  including  the  first  and  the  last 
points,  is  in  lbs.  2.4849,  2.6649,  2.fS391,  2.7081,  2.7726, 
2.8332, 2.8904, 2.9444,  2.9967,  the  common  distance  between 
the  points  is  1  ft ;  find  the  whole  work  done. 

An».  22.0967  foot-ponnda 

69.  A  train  whose  weight  is  100  tons  (including  the 
engine)  is  drawn  by  an  engine  of  160  horse-power,  the  fric- 
tion being  14  \h%  per  ton,  and  all  other  resistances  neglect- 
ed ;  find  the  maximum  speed  whicli  the  engine  is  capable 
of  sustaining  on  a  level  rail.     An».  40^  miles  per  hour. 

7d.  If  the  train  described  in  the  hist  example  bo  movipg, 
tit  a  particular  instant  with  a  velocity  of  16  ^ailes  per  hour. 


EXAMPLES. 


and  the  engin?  workirig  at  full  power,  what  ia  the  accelera- 
tion at  that  instant  ?    (Call  g  =  3*^.)  Ans.  -^^V 

71.  Find  the  horse-power  of  an  engine  required  to  drjig  a 
train  of  100  tone  up  an  incline  of  1  in  60  with  a  velocity  of 
30  miles  an  hour,  the  friction  being  1400  lbs.     . 

Ans.  The  engine  must  be  of  not  lees  than  470f  horse- 
IK)wer.  This  is  somewhat  above  the  power  of  most  locomo- 
tive engines. 

72.  A  train,  of  200  tons  weight,  is  ascending  an  incline 
of  1  in  100  at  the  rate  of  30  miles  per  hour,  the  friction 
being  8  lbs.  per  ton.  The  steam  boiug  shut  off  and  the 
break  applied,  the  train  is  stopped  in  a  quarter  of  a  mile. 
Find  the  weight  of  the  break-van,  the  coefficient  of  fric- 
tion of  iron  on  iron  being  |.  Atis.  11-^  tons. 


is  the  acceltra- 
Ans.  ^^. 

[uired  to  drag  u 
th  a  velocitj'  of 
1.     . 

lan  470|  horse- 
>f  most  locomo- 

ling  an  inclino 
lur,  the  friction 
mt  oflf  and  the 
rter  of  a  mile, 
flicient  of  fric- 
s.  11^  tona. 


CHAPTER    VI. 

MOMENT    OF    INERTIA* 

224.  Moments  of  In6rtia.'-The  quantity  Lmr'  in 
which  m  is  the  mem  of.  an  element  of  a  body,  and  r  its 
ilistance  from  an  axis,  occurs  frequently  in  problems  of 
rotiition,  so  that  it  becomes  necessary  to  consider  it  in 
detail ;  it  is  called  the  moment  of  inertia  of  the  body  about 
the  axis  (Art,  218).  Hence,  "moment  of  inertia"  may  be 
defined  as  follows :  If  the  mass  of  every  particle  of  a  body  be 
multipiied  by  the  square  of  its  distance  from  a  straight  line, 
the  sum  of  the  products  so  formed  is  called  the  Moment  of 
Inertia  of  the  body  about  that  line. 

If  the  mass  of  every  particle  of  a  bjdy  be  multiplied  by 
the  square  of  its  distance  from  a  ^iven  plane  or  from  a 
given  pointy  tiie  sum  of  the  products  so  formed  is  called  the 
moment  of  inertia  of  the  body  with  reference  to  that  plane 
or  that  point. 

If  the  body  bo  referred  to  the  axes  of  x  and  y,  and  if  the 
mam  of  each  parttcio  be  multiplied  by  its  two  co-ordinates, 
.r,  y,  i\>  sum  of  the  products  so  formed  is  called  the 
product  of  inerd-a  of  Ihe  body  about  those  two  axes. 

If  dm  denote  'he  mass  of  an  element,  p  its  distance  from 
the  axis,  and  /  tl>e  moment  of  inertia,  we  have 

•       /  =  l.jMm.  (I) 

If  the  body  Ix?  leforred  to  rectangular  axes,  and  x,  y,  «, 
l)e  the  co-ordinatob  of  any  ''lomont,  then,  according  to  the 
dcfiuitioDs,  the  moments  of  inertia  alM)ut  the  axes  of  x,  y, 
z,  respectively,  will  be 


*  Thit  term  wm  Introduced  b;  Bulor,  knd  bav  now  g<A  into  general  umc  wbeii- 
eror  Kigtd  DjmMnlw  U  iIimUoiI. 


^mmmmmmmmmsmsmm 


mmmmmmmammmmT 


I 


480  EXAMPLES. 

The  momentB  of  inertia  with  respect  to  the  planes  yz,  zx, 
xy  respectively,  are. 


S  vNhn,    S  yWfW,    S  Mm. 


(3) 


The  prodacts  of  inertia  with  respect  to  the  axes  y  and  t, 
K  and  X,  X  and  y,  are 

l,ytdm,    Izxdm,    Xxydm.  (4) 

The  momont  of  inertia  with  respect  to  the  origin  is 

S  (a^  +  y»  +  1^)  rfw  =  1  r*dtH,  (S) 

where  r  is  the  distanoe  of  the  particle  from  the  origin. 

The  moment  of  inertia  of  a  lamina,  when  the  axis  lies  in 
it,  is  called  a  rectangular  moment  of  inertia,  and  when  it  is 
perpendicular  to  the  lamina  it  is  called  a  polar  motnent  of 
inertia,  and  the  corresponding  axis  is  callod  the  rectangular 
or  the  polar  axis. 

The  process  of  finding  momenta  and  produete  of  inertia 
is  merely  that  of  integration  ;  but  after  this  has  been  accom- 
plished for  the  simplest  axes  possible,  they  can  be  found 
without  integration  for  any  other  axes. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  uniform  rod,  of  mass 
m,  and  length  I,  about  an  axis  through  its  centre  at  right 
angles  to  it. 

Lpt  X  he  tne  distanoe  of  any  element  of  the  rod  from  the 
centre,  and  ft  the  mass  of  a  unit  of  length  ;  then  dm  =  ftdx, 
which  in  (1)  gives  for  the  moment  of  inertia  1  itsMx,  or 


-A 


/*:r»<fo, 


JtXAMPLS. 


431 


[a^  +  f^dm,  (2) 
I  the  planes  yc,  zx, 

I.  (3) 

the  axes  y  and  t, 

«.  (4) 

the  origin  is 
dm,  (ft) 

n  the  origin, 
en  the  axis  lies  in 
ia,  and  when  it  is 
polar  motnetU  of 
)d  the  recianffular 

roduett  of  initio 
is  has  been  aooom- 
ley  can  be  foond 


orm  rod,  of  mass 
ts  centre  at  right 

the  rod  from  the 
then  dni  =  ftdx, 
;ia  £  lisMz,  or 


remembering  that  the  symbol  of  summation,  £,  includes 
integration  in  the  cases  wherein  the  body  is  a  continuous 
muss. 
Hence  /  —  ^j^  =  itmP. 

If  the  axis  be  drawn  throngh  one  end  of  the  rod  and 
perpendicular  to  its  length  we  shall  have  for  the  moment 
ol"  inertia 

/  =  \mP. 

2.  Find  the  moment  of  inertia  of  a  rectangular  lamina* 
iihout  an  axis  through  it?  centre,  parallel  to  one  of  its  sides. 

Let  b  and  d  denote  the  breadth  and  depth  rospoctively  of 
the  rectangle,  the  former  being  parallel  to  the  axis.  Im- 
agine the  lamina  composed  of  elementary  strips  of  length  b 
piuullel  to  the  axis.  Let  the  distance  of  one  of  them  fi-om 
ilio  axis  be  y,  and  its  breadth  dy ;  then,  denoting  the  moss 
of  a  unit  of  area  by  ft,  we  have  dm  =  ftbdy,  which  in  (1) 
gives 

If  the  axis  be  drawn  through  one  end  of  the  rectangle,  we 
shall  liave  for  the  moment  of  inertia 

/  =  twd». 

3.  Find  the  moment  of  inertia  of  a  circular  lamina  with 
re8)>ect  to  an  axis  through  its  centre  and  perpendicular  to 
il8  surface. 

Ijet  the  radius  =  a,  and  fi  the  mass  of  a  unit  of  area  as 
before,  then  wo  have 


t/o       ''0 


•  In  all  MM«  mv  shall  anamo  the  thlcknens  of  tho  laminin  nr  plateit  lo  he 
liiaultcalmal. 


..■mamm 
is* 


433 


PARALLSC  AXSS, 


4.  Find  tho  moment  of  inertia  of  a  oircnlar  plate  (1) 
about  u  diameter  as  an  axis,  and  (2)  about  a  tangent. 

Ans.  (1)  imtfi;  (2)  Jma*. 

5.  Find  the  moment  of  inertia  of  a  square  plate,  (1) 
about  an  axis  through  its  centre  and  perpendicular  to  itH 
plane,  (2)  about  an  axis  which  joins  the  middle  points  of 
two  opposite  sides,  und  (3)  about  an  axis  passing  tlirough 
an  angular  point  of  the  plate,  and  perpendicular  to  its 
plane.  Let  a  =  the  side  of  the  plate  and  ^  the  miiss  of  a 
unit  of  area. 


(2)  T»,»irt3;   (3)  |wfl«. 

6.  Find  the  moment  of  inertia  of  an  isosceles  triangular 
plate,  (1)  about  an  axis  through  its  vertex  and  perpen- 
dicular to  its  plane,  and  (2)  about  an  sxis  which  passes 
through  its  vertex  and  bisects  the  base. 

Let  2b  =  the  base  and  a  =  the  altitude,  then 


^0    ''0 


m 


fi  (a^  +  y»)  rfy  dx  =  ^  (3«»  +  b>)  ;  (2)  ^»»i». 


225.  Moments  of  Inertia  relative  to  Parallel 
Axes,  or  Planes. — The  moment  ofimrtia  qfa  body  about 
any  axis  is  equal  foils  moment  of  inertia  abopt  a  parallel 
axis  through  the  centre  of  gravity  of  the  body,  plus  the 
product  of  the  mass  of  the  body  into  the  square  of  the  dis- 
tance between  the  axes. 

Let  the  piano  of  the  paper  pass 
tlirough  tho  centre  of  gravity  of  tho 
body,  and  be  perpendicular  to  tho  two 
parallel  axes,  meeting  them  in  0  and 
O,  and  let  P  be  the  projection  of  any 
element  on  the  plane  of  the  pa])er. 


a  oircalar  plate  (1) 
oat  a  tangs^nt. 
L)  im^;  (2)  |»ia«. 

a  square  plate,  (1) 
perpendicular  to  itn 
ihe  middle  points  of 
sis  passing  tlirough 
)orpeudicuIar  to  its 
and  n  the  mass  of  a 


6    ~  c"  ■ 


I  isosceles  triangular 
vertex  and  pcrpen- 
sxis  which  passes 


ido,  then 


k»  +  «») ;  (2)  ^mb*. 

tive  to   Parallel 

tia  qfa  body  about 
rtia  nbo^t  a  parallel 

the  body,  plus  the 
he  square  of  the  die- 


EH 


BXAMPLES. 


433 


Take  the  centre  of  gravity,  O,  as  origin,  the  fixed  axis 
through  it  perpendicular  to  the  plane  of  thepapw  a»  tlie 
axis  of  z,  and  the  plane  through  this  and  the  pai-allcl  axis 
for  that  of  2*;  and  let  /,  be  the  moment  of  inertia  abont 
tlio  axis  through  G,  /that  for  the  parallel  axis  (hrougn  0, 
a  the  distance,  OG,  between  the  axes,  and  (.r,  //)  any  point, 
/'.    Then  we  shall  have 

/,  =  2  (.i;»  +  yi)  dm ;  /  =  X  [(a;  +  a)^  +  f]  dm. 

Hence        I  —  I^  =  2«  Zxdm  +  a'lrfm  =  a'hti, 

aineo  I.xdin  ~  0,  as  the  centre  of  gravity  is  at  the  origin. 

.-.    /--=  7,  +rt»m,  (1) 

wliich  is  called  the  formula  of  reduction. 

llonco  the  moment  of  inertia  of  a  body  relative  to  anv 
axis  can  be  found  when  tiiat  for  the  parallel  axis  through 
its  centre  of  gravity  is  known. 

Cor.  1. — The  moments  of  inertia  of  a  body  are  iho  same 
for  all  pai-allel  axes  situated  at  the  same  distance  from  its 
centre  of  gravity.  Also,  ot  all  jjarallel  axes,  that  which 
passes  through  the  centre  of  gravity  of  a  body  has  the  least 
moment  of  inertia. 

Cor.  2.— It  is  evident  that  the  same  theorem  holds  if  the 
moments  of  inertia  bo  taken  with  respect  to  i)arallel  planen, 
instead  of  parallel  axes. 

A  similar  proi)erty  also  connects  the  moment  of  iiupiiii 
relative  to  any  point  with  that  relative  to  the  centre  of 
gravity  of  the  body. 

EXAMPLES. 

1.  The  moment  of  inertia  of  a  rectangle*  in  reference 
to  an  axis  through  its  centi-o  and  parallel  to  one  end  is 

*  Sec  Note  to  R«.  t,  Art.  *M ;  strictly  fpt^WiiR,  an  area  bat  s  moment  of  inertta 
no  more  tUau  it  bw  weight. 


■mmxiiniiJnm 


434 


SADIU8  or'  aVBATtOff. 


■^^mtP ;  find  tho  motoent  of  iaertia  in  rdercnee  to  a  parallel 
axis  throngh  one  end. 
From  (1)  we  have 


/,  =  ^m<P  +  -J-  w  =  Jtnrf*.  ^^B 

3.  The  moment  of  inertia  of  an  isosceles  triangle  about 
an  axis  throngh  its  vertex  and  perpendicular  to  its  plane 
is  \m  (3efl  +  i»),  (Art  224,  Ex.  6) ;  find  its  moment  about 
a  i^irallel  axis  through  the  centre. 

From  (1)  we  have 

J  =  ^(3a»  +  i»)  -  ♦a»»»  =  im(V»»  +  J»). 

3.  Find  the  moment  of  inertia  of  a  circle  about  an  axis 
through  its  circumference  and  perpendicular  to  its  p^ine 
(See  Ex.  3,  Art.  224).  Ans.  fnufi. 

4.  Find  the  moment  of  inertia  of  a  square  about  an  axis 
through  the  middle  point  of  one  of  its  sides  and  perpen- 
dicular  to  its  plane  (Ex.  5,  Art  224).  Am.  ^fgtnaK 

226.  Radiiui  of  Qyratic^— Let  k  be  such  a  quantity 
that  the  moment  of  inertia  =  mk?,  then  we  shall  have 


/  =  I,r*dm  =  mk^. 


(1) 


The  distance  *  is  called  the  radius  of  gyraUon  of  the 
body  with  respect  to  the  fixed  axis,  and  it  denotes  the 
distance  from  the  axis  to  that  point  into  which  if  the  whole 
mass  were  concentrated  the  moment  of  inertia  would  not  be 
altered.  The  point  into  which  the  body  might  be  concen- 
trated, without  altering  its  moment  of  inertia,  is  called  the 
centra  of  gyration.  When  the  fixed  axis  passes  through  tho 
centre  of  gravity,  tho  length  k  and  the  point  of  oonoentra- 
tion  are  called  pritmjHtl  radius  and  principal  centre  of 
gyration. 


DQC8  to  a  parallel 


P. 

B  triangle  about 
ular  to  its  plane 
s  moment  about 


e  about  an  axis 

liar  to  its  p^'*ne 

Ans.  |«M^. 

ire  about  an  axis 

rides  and  perpen^ 

Ant.  ^^ftna*. 

such  a  quantity 
e  shall  have 

(1) 

'"  gyration  of  the 
I  it  denotes  the 
rhioh  if  the  whole 
irtia  would  not  be 
might  be  concen- 
rtia,  is  called  the 
lases  through  the 
>int  of  oonoentra- 
ineipal  centre  of 


BADWa  OF  aniATIO/f. 

Let  A:,  =  the  {Nrinoipal  radius  of  gyration  and  r,  the 
distance  of  an  element  fivm  the  axis  through  the  centre  of 
f^ravity;  then  from  (1)  we  have 

=  £  Ti^dm + m<fi,  [by  (I)  of  Art.  225] 

=  mk^^  +  mo*; 

.-.    A*  =  *j»  +  ^»,  (2) 

from  which  it  appears  that  the  principal  radius  of  gyration 
is  Ifie  hast  radius  for  parallel  axes,  which  is  also  evident 
from  Cor,  1,  Art  226. 

Son. — In  homogeneous  bodies,  since  the  mass  of  any  part 
Miiies  directly  as  its  volume,  (1)  may  be  written 


J.f*dV=z  V1^, 


(8) 


where  rfT  denotes  the  element  of  volume,  and  V  the  entire 
volume  of  the  body. 

Hence,  in  homogeneous  bodies,  the  value  of  A  is  inde- 
pendent of  the  density  of  the  body,  and  depends  only  on  its 
form ;  and  in  determining  the  moment  of  inertia,  we  may 
tiike  the  element  of  volume  or  weight  for  the  element  of 
mass,  and  the  total  volume  or  weight  of  the  body  instead 
of  its  mass. 

Also  in  finding  the  moment  of  inertia  of  a  lamina,  since 
k  is  independent  of  the  thickness  of  the  lamina,  we  may 
hike  the  element  of  area  instead  of  the  element  of  mass, 
iind  the  tot^  area  of  the  lamina  instead  of  its  mfw. 

From  (1)  we  have 

*«  =  ^.  (4) 


m 


Similarly, 


*.' 


=  i. 


m 


(6) 


430 


POLAR  MOMENT  OF  lySHT/A. 


henco,  the  square  of  the  radius  of  gyration  with  reaped  to 
any  axis  equals  the  moment  of  inertia  with  respect  to  tha 
same  axis  divided  by  the  mass. 


EXAMPLES. 

1.  Find   the  princij)al   imlins  of  gyration  of  a  straight 
liuc. 

From  Ex.  1,  Art.  224.  wc  have 

therefore  from  (5)  we  have  k\^  =  ^P. 

2.  Find  the  principal  radiu.s  of  gyration  of  a  circle  (1) 
witli  respect  to  a  polar  axis,  and   (2)   with  respect  to  u 


rectangular  axis. 


Ans.  (])|««;  (2)K- 


3.  Find  the  principal  radius  of  gyration  of  a  rectangli! 
with  respect  to  a  rectangular  axis.  Ans.  ^d^. 

4.  Find  the  principal  radius  of  g}'ration  (1)  of  a  square 
with  resiwct  to  a  pohir  axis,  and  (2)  of  an  isosceles  triangle 
with  rcsjtect  to  a  polar  axis. 

Ans.  (1)K;   (2)  ^(^^3  +  ^2). 

227.  Polar  Moment  of  Inertia— If  any  thin  plate,  or 
lamina,  l)e  referred  to  two  rectangular  axes  and  x,  y  he  the? 
co-ordinates  of  any  elemont,  then  (Art.  224)  the  momentw 
of  inertia  about  the  axes  of  x  and  y  respectively,  are  S  yHm 
and  £  x^dm  ;  and  therefore  the  moment  of  inertia  witli 
respect  to  the  axis  drawn  perpendicular  to  the  plane  at  the 
intersection  of  the  axes  of  x  and  y  is 

S  (^  +  y^)  dm. 

Hence  the  polar  moment  of  inertia  of  any  lamina  is  equal 
to  the  sum  of  the  tnomrvts  of  inertia  with  respert  to  any  two 
rectangular  axes,  hjing  in  the  plane  of  the  lamina. 


w 

th 
be 
nu 
re 

III] 
of 

its 
|.h 

itti 


774. 


m  with  respect  to 
vith  respect  to  the 


ktiou  of  a  straight 


tion  of  a  circle  (1) 
with  rcspoot  to  n 

tion  of  a  rectangle 
A  lis.  ^(P. 

m  (1)  of  a  square 
ti  isosceles  triauglo 

(2)  i(i«»  +  *»). 

f  any  thin  plate,  or 
:e9  and  x,  y  he  tho 
334)  the  momenta 
ctively,  are  X  y^dni 
»t  of  inertia  with 
»  the  plane  at  the 


ny  lamina  is  equal 
respert  to  any  two 
1  lamina. 


POI.Ali  MOMENT  OF  TXtlRTIA. 


4;j7 


Coil.— For  every  two  rectangular  axes  in  the  plane  of 
the  lamina,  at  any  jwint,  we  have 

S  xMm  +  2;  i^dm  =  const 

that  is,  the  sum  of  the  moments  of  inertia  with  respect  to  a 
pair  of  rectatigulur  axes  is  constant.  Heucu,  il  one  be  a 
maximum,  the  other  is  a  minimum,  and  vice  versa. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  rectangle  with  respect 
to  an  axis  tlirough  its  centre  and  perpendicular  to  its  plane. 

From  Ex.  2,   Art.    324,   the  rectangular   momenta  of 
inertia  are 

-^^miJP  and  ^»iA» ; 

tlierefore  the  polar  moment  of  inertia  =  ^m  {iP  +  4«) ; 

2.  Find  the  moment  of  inertia  of  an  isosceles  triangle 
,  with  respect  to  an  axis  through  its  centre  parallel  to  its 

1)086,  a  being  the  altitude  and  and  2b  the  base. 

Ans.  ^ina'^;  P  =  -^ciK 

228.  Moment  of  Inertia  of  a  Solid  of  Revolution, 
with  respect  to  its  Geometric  Axia— Let  ihe  axis  be 
that  of  x;  and  let  tho  equation  of  the  generating  curve 
''0  y  =f{x).  Let  the  solid  be  divided  into  an  inlinite 
number  of  circular  plates  perpendicular  to  the  axiii  of 
revolution  ;  let  the  density  be  uniform  and  ft  the  mass  of  a 
unit  of  volume;  and  denote  by  x  tho  distance  of  the  centre 
of  any  circular  plate  from  tho  origin,  y  its  radius,  ami  rfx 
its  thickness ;  thou  the  moment  of  inertia  of  this  circular 
plate  about  an  axis  through  its  centre  and  j)er[K?ndicul!ir  to 
\ia  i>lane,  by  (Ex.  3,  Art.  334),  is 


S^Sa^^S 


4E3SaaswTOiasar 


438 


XXAMPLXS. 


therefore  the  moment  of  inertia  of  the  whole  solid  is ' 

f/[/W^;  (1) 

the  integration  being  takeji  between  proper  limits. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  right  circular  cone 
about  its  axis. 

let  h  =r.  the  height  and  *  =  the  radius  of  the  base ; 
then  the  equation  of  the  generating  curve  is  y  =  *  a-, 
which  in  (1)  gives  for  the  moment  of  inertia, 

10 


m*  Jo    ^  ~ 

=  A'"**,  (since  m  =  ^f^hlA' 


I  since  m 
Therefore    *,«  =  ^. 

2.  Find  the  moment  of  inertia  (1)  of  a  solid  cylinder 
about  its  axis,  b  being  its  radius  and  h  its  height,  and  (2) 
of  a  hollow  cylinder,  b  and  b'  being  the  external  and 
internal  nsdU.  Am.  (1)  f«*» ;  (2)  ^m  {W  +  b^ 

3.  Find  the  moment  of  inertia  of  a  paraboloid  about  its 
axis,  h  being  its  altitude  and  b  the  radius  of  the  base. 

Am,  -g-. 

229.  Moment  of  Inertia  of  a  Solid  of  Revolntion, 
with  respect  to  an  Axis  Perpendicnlar  to  its  Geo- 
metric Aada.— Take  the  origin  at  the  iiitcrscctiou  of  the 


rholo  solid  is 

por  limita. 

right  oircalar  coiic 

dias  of  the  base; 

b 
curve  18  tf  =  T  X, 

irtia, 
0 


>^y 


t  a  solid  cylinder 
ts  height,  and  (2) 
the  external  and 
2)  *»«  (i»  +  b'^. 


raboloid  about  its 
I  of  the  base. 

TTfihb* 
6 


Ant. 


1  of  ReTOlntion, 
Dlar  to  its  Ooo- 

iitcrsc'Ctiou  of  the 


EXAMPLSS. 


489 


axis  of  revolution  with  the  axis  about  which  the  moment 
of  inertia  is  required ;  and  denoting  by  x  the  distance  of 
the  centre  of  any  droular  plate  from  the  origin,  y  its 
radius  and  dx  its  thickness,  we  have  for  the  moment  of 
inertia  of  this  circular  plate,  about  a  diameter,  by  Ex.  4, 
Art  224, 

therefore  (Art  225)  the  moment  of  inertia  of  this  plate 
about  the  pandlel  axis  at  the  dutanoe  x  from  it  is 

therefore  the  moment  of  inertia  of  the  whole  solid  is 

''t'f(l+!f^)^^>  (1) 

the  integration  being  taken  between  proper  limita. 

EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  right  circular  cone 
about  an  axis  through  its  vertex  and  perpendicular  to  its 
own  axis. 

Let  A  =  the  height  and  b  =  the  radius  of  the  base,  then 
the  moment  of  inertia  from  (1) 

2.  Find  the  moment  of  inertia  of  a  cone,  whose  altitude 
=  h,  and  the  radius  of  whose  base  =  b,  about  an  axis 
through  its  ceuire  of  gravity  and  perpendicular  to  its  own 
axis.  Ans.  ^m{h*  +  46»). 


*fci« 


i> 


440 


EXAMPLES. 


3.  Find  the  moment  of  inertia  of  a  pamboloid  of  revolu- 
tion about  an  axis  through  its  vertex  and  periHjndicular  to 
its  own  axis,  the  altitude  being  A  and  the  radius  of  tho 

b 

230.  Moment  of  Inertia  of  Varioua  Solid  Bodies. 


EXAMPLES. 

1.  Find  tho  moment  of  inertia  of  a  rectangular  parallel- 
epiped about  an  axis  through  its  centre  of  gravity  and  par- 
allel to  an  eilge. 

Let  the  edges  be  a,  b,  c;  !»inee  a  parallelopiped  may  bo 
conceived  as  consisting  of  an  infinite  number  of  rectangular 
laminje,  Oaeii  of  which  has  tiie  same  mdius  of  gyration 
relative  to  an  uxi«  ixfrpendicilar  to  its  plane,  it  foUowa 
that  the  radius  of  gynition  of  the  parallelopiped  is  tho 
same  as  that  of  tho  lamina!.  Hence,  the  moments  of 
inertia  roltUivo  to  three  a.\e8  through  the  centre  and  par- 
allel to  the  edges  a,  h,  c.  n^spectively,  are  by  Ex.  1,  Art. 
237,  ^m  (ft»  +  c^),  ^m  («»  -f  r^),  ^,tn  (a*  +  6»). 

2.  Find  the  moment  of  inertia  of  a  rectangular  parallel- 
opiped about  an  edge. 

This  may  be  obtained  immediately  from  the  last  oxam* 
pie  by  using  Art.  225,  or  otherwise  iudefx-ndently  afi 
follows : 

Take  the  thi-ee  edges  a,  b,  c  for  the  axes  of  x,  y,  «, 
ro8i)ectively  ;  let  [t  be  the  mass  of  a  unit  of  volume,  then 
tho  moment  of  inertia  relative  to  the  edge  a  is 

=  r  I  A  (»»+«»)  rf«  rfy ''« 

•/Q    •■  0    '0 


'm- 


iboloiil  of  revolu- 
peri^ndicular  to 
the  radius  of  tho 


0  Sclid  Bodies. 


tanpular  parallel- 
gravity  aud  par- 

lelopijwd  may  bo 
xsr  of  retitangulur 
idius  of  gyration 
plauo,  it  follows 
illelopiped  is  tlio 
the  moments  of 
contro  and  par- 
e  by  Ex.  1,  Art. 

lingular  parallel- 

m  the  last  oxam* 
indofK-ndcntly   as 

axes  of  X,  y,  z, 
of  volume,  then 
a  is 


,'/  'f^ 


c'U 


MOTIOy  OF  IS'ERTIA    OF  A    LAMIXA. 


441 


and  similarly  for  tho  moments  of  inertia  about  the  edLn^s 

/'  and  c.  * ' 

The  moment  of  inertia  of  a  cube  whose  ed^jo  is  a  with 
respect  to  ono  of  its  edges  m  \iia^  =  ^i„u\ 

a.  Find  the  u.jraent  of  inertia  of  a  segment  of  a  -phero 
relative  to  a  diameter  parallel  to  tho  pkue  of  flection  ^ho 
nidms  of  tho  sphere  being  «  and  the  distance  of  tho  lilane 
section  from  the  ct-ntre  b. 

Am.  i^TT  (IGflS  +  \^b  +  0)0^1^  ._  y^), 

231.  Moment  of  Inertia  of  a  Lamina  with  respect 
to  any  Aaris.— When  the  moment  of  inertia  of  a  plane 
fijruro  about  any  axis  is  known,  we  easily  find  tho  moment; 
oL  inertia  about  any  {)arullel  axis  (Art.  225) ;  ,il,wo.  whoa 
the  moments  of  inertia  about  two  rectangular  axes  in  tin- 
j.lano  of  the  fignrc  are  known,  the  moment  of  inertia  about 
i  li(}  straight  line  at  right  angles  to  the  plane  tf  these  axes 
it  their  intersection  is  known  immotliately.  (A^  Ui) ;  wo 
now  proceed  to  find  the  moment  of  inertia  about 'any 
.straight  lino-  in  the  plane  inclined  to  these  axes  at  anv 

JillglO. 

Through  any  point,  0,  as 
■rigin,  draw  two  rectangular 
)>;es,  OX,  OY,  in  the  ])lane  of 


Pig.w 


!he  lamina;  and  draw  any 
iraight  line,  OX,  in  the  plane. 
ii  is  re<piired  to  find  tiie  nio- 
'n  at  of  inertia  about  OX  in 
t.  ima  of  iho  uiomoutp  of  inertia  about  OX  and  OY. 

\A^iP\^  any  point  of  the  lamina...r.  v,  its  n-otimgular, 
.Hid  r,  e,  its  polar  co-ordinates./;  =:  I'M,  and  «  fhe\u,glo 
'OX.  Then  if  f  bo  the  moment  of  iuerllH  of  the  laniiim 
ivlativo  to  O.c,  aand  b  Iho  moments  of  inertia  relative  to 
tlic  axes  of  x  and  ?/  n^speetively,  and  h  tho  product  of 
mertitt  relative  to  the  same  axes,  we  have 


-■.i.-««ry.r>iH^i*^^.^-,-— -Hr^--..---..^:^^-^^^-.^^-. 


^:;^A..:;^;i^Vft..-..-.t/ntYi"--->--'i-'-'i  r-T\i.i  "1 


442 


PRINCIPAL  AXJSS  OT'  A  BODY. 


fil! 


7  =  X  ;)«Jm  =  £  r»  8in«  (0  -  a)  rfm 
=  1  (y  cos  «  —  a;  sin  «)*  rfm 

=  cos*  a  £  y*<i»t  +  sin*  «  £  a^ywi  —  2  sin  «  cos  a  £  aryJw 
=  a  cos*  «  -f  ft  sin'  «  —  2A  sin  «  cos  «.  (1) 

If  wo  choose  the  axes  so  that  the  tenn  h  or  £  xydm  —  0, 
the  expression  for  /  becomes  much  simpler.  The  pair  of 
axes  Bo  chosen  are  called  the  principal  axes  at  the  point ; 
und  ^ho  corresponding  moments  of  inertia  ar(  called  the 
principal  moments  of  inertia  of  the  lamina,  relative  to  the 
point 

If  A  and  B  represent  these  principal  moments  of  inertia, 

(1)  becomes 

7  =  J  co8»  «  +  5  8iii»  «.  (2) 

Hence,  tlie  moment  of  inertia  of  n  lamina  with  respect  to 
any  axis  through  a  point  tnay  be  found  when  the  principal 
moments  with  respect  to  the  point  are  determined. 

232.  Principal  Axeb  of  a  Body.— .1^  any  point  of  a 
rigid  body  and  in  any  plane  there  is  <i  pair  of  principal 
axes. 

Lot  OX,  OY  (Pig.  92),  be  any  rcctiingular  axes  in  tlii^ 
plane ;  let  Ox,  Oy,  Imj  another  set  of  rcctungulur  axes  in 
tli«  same  plane,  inclined  to  the  former  at  an  angle  «;  li't 
a,  ft,  and  h,  as  before,  denote  the  moments  ami  product  of 
inertia  .ibout  OX,  OY,  and  let  {x',  y')  be  any  point,  P, 
referred  to  the  axes  Ox,  Oy.  Then,  using  the  notation  of 
the  last  article,  wo  have 

x'  =  r  cos  (0  —  «) ;    y'  =  r  sin  (9  —  «) ; 

£  x'y'dm  =  j^£  r»  sin  2  {0  —  «)  dm 

=  cos  2«  £  r*  sin  0  cos  d  dm 

—  I  sin  2«  £  r»  (cos*  0  —  sin»  6)  dm. 

Putting  this  =  0,  and  solving  for  k.  wo  nbtain 


toor. 


gin  a  cos  a  £  xi/dm 

(1) 

h  or  X  xydm  =  0, 
pier.  The  pair  of 
%xes  at  the  poiut; 
rtia  ar(  called  Ibo 
na,  relative  to  the 

aoments  of  inertia, 

(2) 

ina  with  respect  to 
when  the  principal 
^rmined. 

—At  any  point  of  a 
t  pair  of  principal 

igular  axes  in  tlie 
cctaiigulur  axes  in 
at  an  angle  « ;  Int 
its  ami  product  of 
be  any  point,  P, 
iig  the  notation  of 

II  (0  -  ft) ; 

-  a)  dm 

'  cos  9  dm 

)b!i  0  —  sin"  0)  dm. 

ol'.taiu 


wfm 


mmmm. 


THRBE   PRINCIPAL  AXXa. 

22  r»  sin  e  COS  (?  d:.n 


tan  2«  = 


£  r»  (cos«  e  —  sin"  1^) 


_  _^2Sa;y^H__  _     2h 
^(^-  y')  dm  -  i^T, 

As  the  tangent  of  an  angle  may  have  any  value,  positive 
or  negative,  from  0  to  co,  it  follows  that  (1)  will  always 
give  a  real  value  for  2«,  so  there  is  always  a  set  of  princi- 
pal axes  ;  that  is,  at  every  point  in  a  body  there  exists  one 
pair  of  rectangular  axes  for  which  the  quantity  h  or 
:^  xy  dm  =  0. 

CoK.— It  may  also  be  shown  that  at  every  point  of  a 
rigid  body  there  are  three  axes  at  right  angles  to  ouo 
another,  for  which  the  products  of  inertia  vanish.* 

♦  Let  a,  6, «,  bo  the  moments  of  Inertia  about  throo 
«xo»,  ox,  OT,  OZ,  at  right  aiigies  to  one  Mother ;  rf,  « 
/,  the  prodncts  of  Inertia  (Smiw,  Xmsr.  \mxy,  rc- 
cixcilvely).  L<tt  Ox  be  any  lino  drawn  ihronjfh  tlio 
.irlf-in,  making  ungles  ,,,  /?,  ,,  wl(|,  the  co  onllualo 
axon. 

Let  Ol,,  IM,  Mf,  he  the  coordinate*  a-,  y,  t,  of  any 
point  P  i,r  the  body  at  wlilca  an  eicnuiit  of  uiaiw  i»  i» 
Hlliiaiod.    Draw  I'N  ixTpendlciiiar  (o  Ox. 

l'rf)jo<ting  tlie  brolion  lino,  GLKP,  on  ON,  (Art. 
lOS),  we  have 

ON  =  *  cog  >i  ^ycotp\■^  ooa  y 

'Ih.  0P»  =  «•  4- »•  +  i •,    and    1  =co»>- a +  «»•/?  + ooa*  y. 

The  rnomert  of  inertU  I  atjont  Ote  =  SmPN" 

-  ttn  (OP»  -  0K») 

=  SW  It'  +  V'  *  »'      (*  cos  «  f  V  com  /f  +  •  eo*  y)«] 

=•  l>»»[(*'+|f'  +  »')(COB«afC0«»^KC0B«  ^)-(irCOB«  +  jrCO«/J+fC<K|y).] 

-  £m  (If*  f  #•)  roB*  a  +  Sm(f «  +  «•)  con*  fi  i- tm  (t«  +  y')  coa'  y 

-  ISmp*  rati  t)  cm  y  —  tSmmt:  coa  ^  co«  a  -  ttm  tmaiiot$ 
=  •  cofi"  o  +  A  co»«  /)<-<•  con"  K  -  W  coa  /?  coa  )- 

-9»co«  >  eosa  -  ycodocoii/?.  (1) 

To  rapreaont  lhi»  Ruometrleally,  take  a  polut  Q  on  ON ;  and  let  Ita  dJatue* 
from O  be  r,  and  Ifcs  co-ordinate*  be*,,  y,,  I,.    Then 

«,  -recMa.    jr,  -  rcuK^,    «.  ^rco*). 


Fig.  814 


■■T--^'it''-^iii'iii'wi».i 


i'-'^ 


h'f: 


444 


TURSS  PRtSCIPAL  AXXS. 


ScH. — In  many  cases  the  position  of  the  principal 
can  bo  seen  at  once.    Suppose,  for  example,  we  wish 
principal  axis  for  a  rectangle  when  the  given  point  is  the 
centre.     Dravr  through  the  centre  straight  lines  parallel  to 
the  sides  of  the  rectangle  ;  then  these  will  be  the  principal 


[Mil  axes         I 
mh  the  I 

f    ia    flw> 


Tbaufor*  (1)  becomef 


I  - 


<w,'  t  ay,'  4  <».'  -1W».«,  -  at>m,  -ytj.y. 


But  the  equation 

dpnoleg  a^vflT^iXfJit'  srivjgit  tavbf-  Is  sU  O;  becanw  a,  ti,  e  are  nccegsBrily  posUlvo, 
eiuee  »  mr-  I  at  lovrtift  U  MWfitlally  positivu,  being  the  sain  u(  •  bobiIwi'  of 
Miuarr  1  ({  U  tt  point  on  tbU  uUlpcoid,  (3)  Ut'oomee 


I  c  SmPn*  <c 


!i 


or  thn  moment  of  in«rtift  »bont  tny  line  thruugh  O,  h  mcMnred  by  the  sqaut;  of 
tho  roctiiruckl  of  thfl  radlun  Tector  of  tUi»  elll[>M>id,  which  coloeidos  wllh  the 
liuo. 

'{Ilia  li  ckUed  Uiit  niommtat  cS^/mld,  and  ww  first  aaed  by  Oauchy,  Mrtrcimt  d» 
Hath.,  Vol,  n.  It  tuti  no  phyxical  oxlcteDoe,  hat  In  an  artiRoi  to  bring  under  Ute 
rauthodH  of  (jeoinctry  tho  pro|)ert5en  of  inomcTiln  of  tiioKla.  The  mouienlal  elllp. 
gdid  han  a  drtlulte  form  tor  cvory  polBt  of  a  rigid  body. 

Now  every  elli|if>old  hoa  thvoe  axoc^,  to  which  If  il  \*  ivlbrred,  the  coefficients  of 

y»,  cr,  xy  vanlidi,  aud   therefore  (!)),  when  tnuiBfbriued  to  those  asus  takes  the 

form  .  _ 

A/,'  (  By,"  \  0,«  =  1;  («) 


Biul  hence  (1)  or  (»)  wh'in  raAnred  to  these  axcB,  becoroee 
1 «  A  eoe*  •  +  B  to**  ^  +  0  COB" »-, 


m 


whoro  A,  B,  C,  are  Uie  niomentti  of  inertia  of  the  body  about  then!  a:  m. 

When  three  recxangalar  axe;!,  meeting  In  a  t^veu  point,  arc  chuRon  w)  that  ihe 
prodncM  of  Inertia  all  v«Hl»h,  tlie>  are  c*lle<l  the  f)riiteip3l  epsu  at  the  given 
piiliil. 

Tht)  Ihnw^  planen  thtoogh  any  two  priticlpal  axM  iir:>  eallw'  itio  prindija/  }>!cme^ 
at  lltt'  ^Ivon  point. 

The  monionts  of  inertia  ahont  lh«  piincjial  nxwi  at  o«iy  point  are  railed  the  prin- 
fi;in/  tncnumtJi  tf  i»*rHa  at  thai  p«ilnt. 

If  the  (hreo  prlnii|)al  moinenti>  of  Inertia  ofa  body  are  Mjniil  to  one  another,  the 
oUl|iHoi'l  '4)  I«comeii  a  Hp'jere,  ulni^c  A  =.  B  -  V ;  3nJ  therofotc  the  inumeut  of 
iMrlla  about  every  other  atlt  it  oqiuU  to  L-nae,  tor  (R)  becoiuua 

1  ■-'<  A  <oot' •  ♦  cm*  <$  4^  ow<  i)  o"  A ; 

and  every  <uti»  In  a  pru»lp«l  Ml«.    (See  IbiUth'g  UI>rii]  Dynamic*,  p.  I«,  Price's 
AuftV  S.cliv  Vui,  a.  1.  IW,  I-irWrJUglilDyuaiulcc,  p  TO,  iie  - 


# 


principal  oxoa 

1,  we  wish  tlio 
a  point  is  tliu 
11C8  parallel  to 
I  tbo  priuci(Ml 


*. 


=  I. 


m 


(8) 


9i'0a««riiy  postllro, 
un  uf  t  DDiDlter  of 


td  by  the  aqnsro  of 
coloeldM  with  the 

^auchy,  Mtereitet  d» 
I  to  bring  under  tlie 
lie  mouienlal  I'lliji- 

Ihc  cocfllcleiit«  of 
boDe  iLYuii  takuD  tlio 

(*) 


(») 

iP.ni  a: :m. 

churan  no  that  the 
aiM  at  the  givvu 

ilio  prlneiiMtl  planti 

tarct-«nei1  IhopHn- 

lo  on«  another,  tbo 
'ore  (ho  inomeut  of 


laniic*,  It.  1«.  IMco's 


TBRSE  PBTNCIPAI,   AXES. 


««8 


axes;  becaase  for  orery  element,  dm,  on  one  side  of  the 
axis  of  X  at  the  point  («,  y),  there  is  another  elenient  of 
equal  mass  on  the  other  side  at  the  point  {x,  —y).  Hence, 
yLxydm  consists  of  terms  which  may  bo  arranged  in  pairs, 
so  that  the  two  terms  in  a  pair  arc  numerically  equal  but 
of  opposite  signs ;  and  therefore  £  xy  din  —  0. 

Agiiin,  if  in  any  uniform  body  a  straight  lino  can  Imj 
drawn  with  respect  to  which  the  body  is  exactly  symmetri- 
cal, this  must  be  a  principal  axis  at  every  point  in  its 
length.  Any  diameter  of  a  uniform  circle  or  sphere  or  the 
axis  of  a  parabola  or  ellipse  or  hyperbola  is  a  principal  axis 
at  any  point  in  its  line;  but  the  diagonal  of  a  rectangular 
])lat«  is  not  for  this  reason  a  principal  axis  at  its  middio 
point,  for  every  straight  line  drawn  perpendicular  to  it  is 
not  equally  divided  by  it. 

Let  the  body  be  symmetrical  about  the  plane  of  xy,  then 
for  every  element  dm^  on  one  side  of  the  plane  a*:  the  point 
{x,  y,  «),  tliero  is  another  elem  nt  of  equal  mjusa  on  the 
other  side  at  the  point  {x,  y,  — *).  Hence,  for  such  a  body 
2  xz  dm  =  0  and  £  yz  dm  ■=.  0.  If  the  body  bo  a  lamina  in 
the  plane  of  xy,  then  z  of  every  element  is  zero,  and  wo 
have  again  £  xz  dm  =  0,  !•  yt  dm  =  0. 

Thufi,  in  tlic  case  of  the  ollipsoid,  the  three  principal 
sections  are  all  planet  of  symmetry,  and  therefore  the  three 
axes  of  (he  ellipsoid  are  principal  axes.  Also,  at  every 
jKiint  in  a  lamina  one  principal  axis  ia  the  pcrpendioular  to 
tlie  pltme  of  the  lamina, 


EXAMPLES. 

1.  Find  the  moment  of  inertia  of  a  rectangular  lamina 
:il)uut  a  diagonal. 

From  Ex.  2,  Art.  5S24,  the  moments  of  inertia  about  two 
linos  through  the  centre  parallel  to  the  sides  (principal 
moments  of  inertia)  ore 


fm 


446 


BXAMPLSa. 


•  » 


I 


^m<2*    and    VfWii*; 

■whoro  h  ami  rf  aro  the  breadth  and  dopth  respectively. 

Also,  if  «  be  the  angle  which  the  diagonal  raakee  with 
the  tiide  i,  we  have 


«n»  a  = 


I>>  +  (P' 


00^  rt  = 


d2  +  <^ 


SnVwtituting  these  values  for  A,  B,  sin*  «,  cos»  «,  in  (2)  of 
Art.  231,  wo  have 


=  ^m 


2.  Find  the  moment  of  inertia  of  an  isosceles  trfangnlar 
plat«  about  an  axis  through  its  centre  and  inclined  at  an 
angle  «  to  its  axis  of  syrametry,  a  being  its  altitude  and  2ft 
its  baae.  Ans.  ^  (^a*  cos*  «  H-  J»  sin*  «). 

3,  Find  the  moment  of  inertia  of  a  square  plate  about  a 
diagonal,  «  being  a  sidi!  of  the  square.  Jns.  ^maK 

233.  ProductB  of  Inertia.— The  value  of  the  product 
of  inertia  at  any  iwint  may  be  made  to  depend  on  the  value 
of  the  product  of  inertia  for  parallel  axes  through  the  cen- 
tre of  gravity.  Let  {x,  y)  lie  the  position  of  any  element^ 
dm,  referred  to  axes  through  any  a88ign<'d  point ;  {x',  y')  the 
jwsition  of  the  element  referred  to  parallel  ixes  through 
the  centre  of  gravity,  and  (A,  k)  the  centre  of  gravity 
referr(Hl  to  the  firat  pair  of  axea     Then 

ar  =  «'  +  A,    y  =.  y'  +  h) 

tlM»fore      "£  xydm^I.  {x'  +  h)  {y'  +  h)  dm 

=  Y,r!y'  dm  +  hk^dm,  (1) 


since  S  wwf'  =  0,  and  S  my' 


0. 


J,#Sv 


SXAMPLSA 


U1 


KJCfcively. 

U  makes  with 


a«  «,  in  (2)  of 


elea  triangular 
inclined  at  an 
Ititude  and  2b 
H-  J»  8in»  «). 

plate  about  a 
Ans.  '^ftnaK 

of  the  prodnct 
id  on  the  value 
rough  the  cen- 
f  any  element, 
iut ;  {x',  tj')  the 
il  ixes  through 
itre  of  gravity 


Im 


(1) 


ScH.— By  (1)  we  may  often  find  the  product  of  inartia 
for  an  assigned  origin  and  axes.  Thus,  suppose  we  roqnire 
the  product  of  inertia  in  the  case  of  a  rectangle,  when  the 
origin  is  at  the  comer,  and  the  axes  are  the  edges  which 
meet  at  that  ooruer.  By  Art  332,  Sch.  we  have  I.z'y'din 
=  0;  therefore  from  (1)  we  have 

and  as  h  and  k  are  known,  being  half  the  longthn  of  the 
edges  of  the  rectangle  to  which  they  are  rcsiiectivcly 
parallel,  the  product  of  inertia  is  known. 

BXAM  PL.BS. 

Find  the  expressions  for  the  moments  of  in«tia  in  the 
following,  tiie  hwiies  beiag  wipposed  homogeoeous  in  all 
cases. 

1.  The  moment  of  inwtia  of  a  rod  of  length  a,  with 

respect  to  an  axis  perpendicular  to  tite  rod  and  at  a  distance 

d  from  ita  middle  point  ,  /a"         \ 

*  Ans.  m{^~  +  dn- 

2.  Tlie  moment  of  inertia  of  an  arc  of  a  circle  whose 
radius  is  a  and  which  subtends  an  angle  2«  at  the  centre,  (1) 
about  an  axis  through  its  centro  perpendicular  to  its  plane, 
(2)  about  an  avis  through  its  middle  point  perpendicular  to 
its  plane,  (3)  about  the  diameter  which  bisects  the  arc. 

Ans.  (l)«,a-  (2)  2„.  (!_?!£_«)«.;  (8)  «, (l -.!l|^ «*. 

3.  The  moment  of  inertia  of  the  arc  of  a  complete 
cycloid  whose  length  is  a  with  respect  to  its  base. 

Ans.  ^ffna', 

4.  The  moment  o(  inertia  of  an  equilateral  triai.gle,  of 
aide  a,  relative  to  a  line  in  its  plane,  parallel  to  a  side,  at 
the  disUutoo  d  twiu  its  oeuke  of  gravity.  . ' 

Am.  m  (I  +  <p). 


riM 


di«..^».i■>..W.t.■l.,.L.-^:.ljiA^t>fc•^.^^|J.^,■^.,,,  .y:...r..^^-....^-^-    . 


418 


SXAMPLSS. 


h 


6.  Oivcn  a  triangle  whoBc-  sides  are  a,  b,  e,  and  whoso 
jwrpendiculars  on  these  sides,  from  the  opposite  vortices 
are  p,  q,  r,  respectively ;  find  the  moment  uf  inertia  of  the 
triangle  about  a  line  drawn  through  each  vertex  and 
parallel  respectively,  (1)  to  the  side  a,  (2)  to  the  side  b,  {?,) 
to  the  side  e.  Arts.  (1)  Impf^;  (3)  |tw7« ;  (3)  ^m,^, 

C.  Find  the  moiTient  of  inertia  of  the  h'iiinglo  in  the  last 
example  relative  to  the  three  lines  drawn  through  the 
centre  of  gravity  of  the  triangle  and  parallil  respectively 
to  the  sides  a,  b,  c.  Ann.  ^mf\  if,m(j*\  i-xiniK 

T.  Find  the  moment  of  inertia  of  the  triangle  iu  Ex.  5, 
relative  to  the  three  sides  a,  h.  c,  respectively. 

Am.  {nlf^'f  ifUKf;  \mr'K 

8.  The  monioiit  of  inertia  of  a  right  angled  triangle^  oi 
hvputlienose  c,  relative  to  a  perpendicular  to  its  plane 
J)a8>itig  through  the  right  augle.  Av.^.  \m^, 

!t.  Tiic  moment  of  inertia  of  n  ring  vvhcisc  outer  atttf 
inner  radii  aie  a  and  b  respectn ely,  (1)  with  respect  lo  a 
IM)lar  axis  tiiroiigh  its  centre,  and  (2)  with  respect  to  a 
diameter.  Am.   (1)  ^m  («»  -|-  h^) ;  {%)  \m  («»  +  S^). 

10.  The  mof'ienf  of  inertia  of  an  ellipse,  (1)  with  respect 
to  ')\f>  miijor  axis,  (i)  with  respect  to  its  mhior  axis,  and  (3) 
with  reaiiet  to  an  tt>«iH  through  its  centre  and  pc-riiendieular 
U)  it«  plane. 

4n».    (\)  ^110 \   |l)  Imn'i;    (.1)  \m  («»  +  A«). 

11.  The  mmmui  uf  inertia  of  Hie  surface  of  a  rfphero  of 
radius  a  ahout  it.s  diameter.  Aiis.  ImaK 

li.  The  moment  of  inertia  of  a  ri(i)i(  pri«m  whose  haae 
is  !i  rigiit  anglal  triangle,  witli  l'OB[wot  to  au  axis  passing 
{limutrh  thr  eentres  of  gravity  of  the  eud%  tUo  tldoa  cou- 
laining  the  right  angle  of  the  triangular  base  being  a  and  b 
and  the  height  of  the  prism  c.  Ans.  ^m{a^+¥). 


ifeite 


MMMMMM 


',  c,  and  whoso 
jposite  vortices 
L  inertia  of  tho 
3h  vertex  and 
tlie  side  h,  (3) 
r« ;  (3)  \m,^. 

iglo  in  the  last 
1  through  the 
It'l  respectively 

angle  in  Ex.  5, 

r. 

\  /«'/' ;  f/Hr*. 

led  ihimgh,  of 
'    to    its    phe 

u.-ir  outer  s»j«t 
h  respect  to  a 
I  respect  to  a 
[m  (rt»  +  HI). 

I)  witli  rosjwct 

»r  axis,  and  (3) 

ptriiendicuUir 

ni  {(ii  +  *«). 

I  if  (I  riphore  of 
Alts.  |i//«^, 

(III  whoge  baao 
lu  avis  pasHtng 
tUo  tidofl  oou- 
being  a  and  b 


XXAMPLJB& 


M» 


13.  The  m(»D(«t  of  ioertia  of  a  right  pmnt  vhoae  height 
is  c,  about  an  axis  passing  throngh  the  centres  of  gravity  of 
the  ends,  tho  base  of  the  piistQ  being  an  isosceles  triangle 
whose  base  is  a  and  height  b.  /a>      Nt\ 

■^-id  +  r)- 

14.  The  moment  of  inertia  of  »  sphere  <  f.  radins  a,  (1) 
relative  to  a  diameter,  and  (3)  relative  to  a  tangent. 

Ans.  (l)fwo»;  (3)  |»ja». 

15.  Tho  moment  of  inertia,  about  its  axis  of  rotation,  (1) 
of  a  prolate  spheroid,  and  (3)  of  an  oblate  spheroid. 

Ans.  (l)|«ift»;  (3)|7»fl2. 

16.  Tho  moment  of  inertia  of  a  cylinder,  relative  to  an 
axis  perpendicular  to  its  own  axis  and  intersecting  it,  (1)  at 
a  distance  c  from  its  end,  (3)  at  the  end  of  the  axis,  and  (3) 
■.hi  f  ho  middle  point  of  the  axis,  the  altitude  of  the  cylinder 
boujg  fi  and  radius  of  its  base  a. 

Ans.  (1)  ima»  +  ^m  (A»  +  8Ac  +  c»)  ; 

(3)  ^m  (3«»  +  4/t«) ;  (3)  ^m  (7*3  +  3«») 

moment  of  inortirt  of  an  ollipso  about  a  central 
radius  vector  r,  making  an  angle  «  with  the  major-axlB. 

Am.  im-^. 

18.  The  moment  of  inertia  of  the  area  of  a  parabola  cut 
off  by  any  ordinate  at  a  distance  a,  from  the  vertex,  (1) 
aliout  the  tangent  at  tho  vertex,  and  (3)  about  the  axis  of 
tho  iMuiibola. 

Ans.  fwa? ;  (3)  ^my^  where  y  is  the  ordinate  correspond- 
ing to  «. 

l».  The  moment  of  inertia  of  the  area  of  the  lomniscatey 
»■»  =z  «'  COS  m,  about  a  lini*  through  the  origin  in  its  plane 
uiid  periRiudicular  to  it^  a^is.  Am.  ^ftn  (3rr  f  8)  «». 


wg,?ii  iwwgigflFgy 


*■ 


■■immii  fin  fiifrfrrtiii 


460  MXAMP/.ES. 

20,  The  moment  of  inertia  of  the  ellipfsoid, 

^  +  ^  +  ^-^- 

about  the  axis  a,  ft,  c,  respectively. 

Am.  (1)  im(*»  +  c») ;  (2)  i«i  (c»  +  a») ; 
(3)  iw  (a»  +  J»). 


j^  s 


1, 


J«i  (c»  +  a») ; 


''s^rtliBSUBSSH 


CHAPTER    VII. 

ROTATORY     MOTION. 

234.  Impressed  and  Infective  Forces.— All     wes 

acting  on  a  body  other  than  tho  mutual  actionft  ,  tho 
particles,  are  called  the  Impressed  Forces  that  act  on  the 
body. 

ThuB,  when  a  ball  is  thrown  in  vacuo,  the  impresst*! 
force  is  gravity;  if  a  ball  is  rotating  about  a  vertical  axis, 
the  impressed  forces  are  gravity  and  the  reaction  of  tho 
axis. 

The  impnssod  or  external  forces  are  the  cause  of  the  motion  and  of 
all  tlio  other  forces.  Which  are  the  impraosed  forces  depends  npon 
the  particular  system  which  is  under  consideration.  Tho  same  force 
may  be  external  to  one  system  and  inn.  rnal  to  another.  Thus,  tho 
pressure  between  the  foot  of  a  man  and  tlie  deck  of  a  ship  on  which 
he  ifl,  is  external  to  tho  ship  and  also  to  the  man  and  is  thfl  cause  of 
his  own  forward  motion  and  of  a  slight  backward  motion  o,  Jio  ship ; 
but  if  the  man  and  ship  are  considered  as  parts  of  one  system  the 
pressure  is  internal. 

When  a  particle  is  moving  as  part  of  a  rigid  body,  it  is 
acted  on  by  the  external  impressed  forces  and  also  by  the 
molecular  reactions  of  the  other  particles.  Now  if  ".his 
particle  were  considered  as  separated  from  the  rest  of  the 
body,  and  all  the  forces  removed,  there  is  some  one  force 
which,  singly,  would  move  it  in  the  same  way  as  before. 
This  force  is  called  the  Effective  Force  on  the  particle;  it  is 
evidently  the  rtsultant  of  the  impressed  and  molecular 
forces  on  the  particle. 

Thus,  the  effective  force  is  that  part  of  the  impressed  force  which 
is  effective  in  causing  actual  motion.  It  is  tho  force  which  is  required' 
for  prodncing  tUo  deviation  from  the  straight  line  and  the  change  of 


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^.  nil  i«riijt;iMiii|»iiirn;TiiniiirwMir(i 


452 


D  'A  LEMnKR  T  'S  PRINCIPLE. 


vplocitv.  If  a  particle  is  revolving  with  constant  velocity  round  a 
fixed  axis,  the  effective  force  is  the  ceatripetal  forc«  (Art.  198).  If  a 
heavy  Ixxly  fulls  wiihoiit  rotation,  the  whole  force  of  gravity  is 
effective  ;  but  if  it  \»  rotating  about  a  liorizoutal  axis  the  weight  goeti 
partly  to  kiJauce  the  prcHsure  on  the  axis. 

If  we  suppose  tlie  particle  of  mass  m  to  bo  at  the  point 
(r,  y.  z)  at  niiy  time,  /,  flnd  resolve  the  forces  acting  on  it 
into  the  tiireo  axial  com])onent8,  X,  Y,  Z,  the  motion  may 
be  found  [Art.  108  (2)]  by  solving  the  Bimultj.neou8  equa- 
tions 


(Pi/ 


=  r 


m  ^,  :=  Z. 


a) 


■^«  ite- 


If  we  regard  a  rigid  body  as  one  in  which  the  particles 
retain  inviiriable  po-sitions  with  respect  lO  one  another,  so 
that  no  external  force  can  alter  thon  (Art  43),  we  might 
write  down  the  equations  of  the  several  particles  in  accord- 
ance with  (1),  ii"  all  the  forces  wore  known.  Such,  how- 
ever, is  not  the  ease.  We  know  nothing  of  ihe  mutual 
actions  of  the  particles,  and  ccn80<pjently  cannot  determine 
the  motion  of  tlio  bodv  by  calculating  the  motion  of  its 
particles  separately.  ^\  hen  there  are  several  rigid  bodies 
which  mtitually  act  and  react  on  one  another  the  problem 
becomes  still  more  comidicated. 


235.  D'Alombert'B  Principle,  •—By  D'Alembert's 
Principle,  however,  all  the  necessary  equations  may  be 
ol)tained  without  writing  down  the  equations  of  motion  of 
llio  several  iwirHoles.  and  witnout  any  assumption  as  to  the 
nalureof  the  mutual  actions  except  the  following,  which 
may  l»e  i-egardxHi  as  a  natural  oonsoqueitco  of  the  laws  of 
motion. 

TfiK  infernnl  nrtiorm  and  rcaciions  of  any  ayntnn  of  rigid 
bodies  in  motion  arfi  in  (fiuiUbrinin  nmonff  fhemsehvu. 


*  latrodaccd  »>y  y'Alenibcrt  lii  n«. 


D 'ALEMBERT'S  PRI^CtPLB. 


458 


velocity  round  a 
(Art.  198).  If  a 
CO  of  gravity  is 
the  weigbt  goes 


I  at  the  point 
J  acting  on  it 
e  motion  may 
ituneous  cqvia- 

\  the  particlcd 
)ne  another,  so 
43),  we  might 
cles  in  accord- 
I.  Such,  hew- 
)f  the  mutual 
inot  determine 
!  motion  of  its 
111  rigid  bodies 
or  tlie  problem 


D'Alombert's 
fttions  may  be 
18  of  motion  of 
ption  a8  to  the 
•llowing.  which 
of  the  laws  of 

system  of  rigid 
■temselws. 


\ 


The  axial  accelerations  of  the  particle  of  mass  m,  whicli 

cPx    dt^v    (Pz 
is  moving  as  part  of  a  rigid    body,  ai-e    ^,  -^i^  -^^' 

Let/  be  their  resultant,  then  the  eflfeotive  force  is  measured 
by  mf  Let  ^and  R  be  the  resultants  of  the  impressed  and 
molecular  forces,  i-espectively,  on  the  particle.  Then  mf 
is  the  resultant  of  F  and  R.  Hence  if  mf  bo  reversed,  tlio 
three  foi-ces,  F,  R,  and  mf,  are  in  equihbrium. 

The  same  reasoning  may  be  applied  to  every  particle  of 
eiich  body  of  the  system,  thus  furnishing  three  groups  of 
forces,  similar,  respectively,  to  F,  E,  and  mf:  and  these 
three  groups  will  form  a  systeir  of  forces  in  equilibrium. 
Now  by  D*Alembert's  principle  the  group  7^  will  itself 
form  a  system  of  forces  in  equilibrium.  Whence  it  follows 
that  the  group  F  will  be  in  equilibrium  with  the  group  mf. 
Hence, 

If  forces  equal  and  exactly  opposite  to  the  effective  ^forces 
were  applied  at  each  parfick  of  the  system,  they  would  l>e 
in  equilibrium  with  the  impressed  forces. 

That  is,  H'A'emhcrCs  principle  asserts  that  the  lohoU 
effective  forces  of  a  system  are  together  eqnivaknc  to  the 
impressed  forces. 

SoH. — By  this  principle  the  solution  of  a  problem  in 
Kinetics  is  reduced  to  a  problem  in  Statics  ha  follows :  We 
first  choose  ihe  co-ordinates  by  means  of  which  the  position 
of  the  system  in  space  may  be  fixwi.  Wo  then  express  the 
effecii^o  forces  on  each  element  in  terms  of  its  cc-oi'dinates. 
Those  effective  forces,  reversed,  will  be  in  equilibrium 
with  the  given  impressed  forces.  Lastly,  the  equations  of 
motion  for  each  body  may  b^  formed,  as  is  usually  done  in 
Statics,  by  resolving  in  three  directions  and  taking  mo- 
ments abont-  throe  straight  linos.  (See  llouth's  Rigid 
Dynamics,  Pirio's  Rigid  DynamicB,  Pratt's  Mech's,  Price's 
Anal.  Mech's,  Vol.  IL) 


^.^•i^ 


^HMMWKMMMWMM 


464 


ROTATION  OF  A  liWID  BODY. 


236.  Rotation  of  a  Rigid  Body  about  a  Fixed 
ii^B  under  the  Action  of  any  Forces.— Let  any 
plaue  pagRing  through  the  axis  of  rotation  und  fixed  in 
space  be  taken  as  a  piano  of  reference.  Let  m  be  the  mass 
of  any  element  of  the  body,  r  its  distance  from  the  axis, 
and  0  the  angle  which  a  pkuo  through  the  axis  and  the 
element  makes  with  the  plane  of  rekronce. 

Then  the  velocity  of  m  in  a  direction  yrpendicular  to 

the  plane  containing  the  element  and  the  axis  is  r-^-. 

The  moment  of  the   momenluni*  of   this  particle  about 

the  axis  ia  mr'  --,-•    Hence  the  moment  of  the  momenta  of 
at 

all  the  particles  is 


d6 

at 


0) 


Since  the  particles  of  the  body  are  rigidly  connected, 

it  is  clear  that  -=j  is  the  same  for  every  particle,  and  is  the 

angular  velocity  of  the  body.     Hence  the  moment  of  the 
momenta  of  all  the  particles  of  the  body  about  the  axis  is  the 
moment  of  in^tia  of  the  body  about  the  axis  multiplied  by 
the  angular  velocity. 
The  acceleration  of  m  perpendicular  to  the  direction  in 

whicii  r  is  measured  is  r  -^ ,  and  therefore  the  moment  of 

the  moving  forces  of  m  about  the  axis  is  'wr*-,^-    Hence, 

the  moment  of  fhe  effective  forces  of  all  the  particles  of  the 
body  about  the  axis  is 


Iwr» 


(m 


(2) 


which  is  the  moment  of  inertia  of  the  body  about  tJie  axis 
mkltiplied  by  the  angular  acceleration. 

*  Called  alHi)  Angular  Momentum.    (Soo  Itrlo'ii  Rlcld  Dy niunlce,  p.  44.)    . 


>ut  a  Fixed 

ses.— Let  any 
1  and  fixed  in 
m  be  the  mass 
from  the  axis, 

0  axis  and  the 

^rpendicular  to 

.    .       do 
LC  axis  \a  ''j-.' 

particle   about 

;he  momenta  of 

(1) 

idly  connected, 

icle,  and  is  the 

moment  of  the 
t  the  axis  is  the 
is  multiplied  by 

the  direction  in 
the  moment  of 

fir^-jp-    Hence, 

1  particles  of  the 

dy  about  the  axis 

DyniunlcB,  p.  44.) 


ROTATION  OF  A   RIGID  BODY. 


455 


(1)  Let  the  forces  be  impulsive  (Art.  203) ;  let  6),  w',  be 
the  angular  velocities  just  before  and  just  after  the  acaon 
of  the  forces,  and  N  the  moment  of  the  impressed  forces 
about  the  axis  of  rotation,  by  which  the  motion  is  pro- 
duced. 

Then,  since  by  D'Alembert's  principle  the  effective 
forces  when  reversed  are  m  equilibrium  with  the  impressed 
forces,  we  have  from  (1) 

w'  S  mi*  —  w  X  rnr*  =  JV"; 


« 


(J  := 


N 
"Lmt* 


'  moment  of  impulse  about  axis , 
~  momo^IofTncrtia    about  axis ' 


(8) 


that  is,  the  change  in  the  ai^^^^r  velocity  of  a  body,  pro- 
duced byan  impulse,  is  equal  to  the  m^^ient  of  the  impulse 
divided  by  the  moment  of  inertia  of  the  body. 

(2)  Let  the  forces  be  finite.      Then  taking  tiicr.'^^nts 
about  the  axis  as  before,  we  have  from  (2) 


(Pd 
dt^ 


JL 

^mi^ 


moment  of  forces  about  axis . 
~  moment  of  inertia  about  axis ' 


(4) 


that  is,  tU  angular  acceleration  of  a  body,  produced  by  a 
force,  is  equal  to  the  mommt  of  the  force  divided  by  the 
moment  of  inertia  of  the  body. 

By  integrating  (4)  we  shall  know  the  angle  through 
which  the  body  has  revolved  in  a  given  time.  Two  arbi- 
trary constants  will  appear  in  the  integrations,  whose 
values  are  to  be  determined  from  the  given  initial  value? 

of  0  and  ~-    Thus  the  whole  motion  can  bo  L\mA,  and 


456 


MXAMPLE. 


WO  shall  conseqaontly  be  able  to  determine  the  position  of 
the  body  at  any  instant 

ScH. — It  appears  from  (3)  and  (4)  that  the  motion 
of  a  rigid  body  round  a  fixed  axis,  under  the  action  of  any 
forces,  depends  on  (1)  the  moment  of  the  forces  about 
that  axis,  and  (2)  the  moment  of  inertia  of  the  body  about 
the  axis.  If  the  whole  mass  of  the  body  were  concentrated 
into  its  centre  of  gyration  (Art.  226),  and  attached  to  the 
fixed  axis  of  rotation  by  a  rod  without  mafls,  whose  length 
is  the  radius  of  gyration,  and  if  this  system  were  acted  on 
by  forcoe  having  the  same  moment  as  before,  and  were  set 
in  motion  with  the  same  initial  values  of  6  and  the  angular 
velocity,  then  the  whole  subsequent  angular  motion  of  the 
rod  would  be  the  same  as  that  of  the  body.  Hence,  we  may 
say  br'ofly,  that  a  body  turning  about  a  fixed  axis  is 
Irinetically  given  when  its  maaa  ana  radius  of  gyration  are 
known. 

EX  ADA?  L  F. 

A  rough  circular  horizontal  boi.  i  is  capable  o£  vixoh']ng 
freely  round  a  vertical  axis  through  its  centre.  A  man 
walks  on  and  round  at  the  edge  of  the  board;  when  he 
has  completed  the  circuit  what  will  be  his  position  in 
space  ? 

Let  a  be  the  radius  of  the  board,  M  and  M'  the  maeses 
of  the  board  and  man  respectively ;  B  and  0'  the  angles 
described  by  the  board  and  man,  and  i^^the  action  between 
the  feet  of  the  man  and  the  board. 

The  equation  of  motion  of  the  board  by  (4)  is 

Fa  -  ^*.»^' 

(  Since  the  action  between  the  man  and  the  board  is  con- 
tinually tangent  to  the  path  described  by  the  man,  the 
equation  of  motion  of  the  man  is,  by  (5)  of  Art.  20, 


mmmm* 


the  position  of 

tat   the  motion 

le  action  of  any 

be  forces  about 

the  body  about 

ere  concentrated 

attached  to  the 

!8,  whose  length 

oa  weie  acted  on 

re,  and  were  set 

and  the  angular 

ar  motion  of  the 

Hence,  we  may 

a  fixed  axis  is 

I  of  gyration  are 

able  o£  Vixohing 
centre.  A  man 
board;  when  he 
I  his  position  in 

id  M'  the  masses 
and  0'  the  angles 
e  action  between 

(4)i8 


the  board  is  con- 
by  the  man,  the 
,f  Art.  «0, 


"'^wnmmP'- 


m!m»!»s^mmmmm>efmBii>i^>m^f>f»»!" 


TBI!  COMPOUND  VENDULUM. 


IP         V    ^ 


m 


Eliminating  i^and  integrating  twice,  the  constant  being 
zero  in  both  cases,  because  the  man  and  board  start  from 
rest,  wo  get 

Mk?0  -  M'aW.  (1) 

When  the  man  has  completed  the  circuit  we  have  0  +  0' 
=  %iT\  also  ^'i*  =  -.    Substituting  these  in  (1)  we  get 


V 


*  — 


2nM 


-iM'  +  M' 


which  giTOs  the  angle  in  space  described  by  the  man. 
If  M  —  M',  this  becomes 


and 


d'  =  \ir; 
0  =  f7r, 


which  is  the  angle  in  space  described  by  the  board.     (See 
Rye^b's  Rigid  Dynamics,  p.  G7.) 

237.  The  Compound  Pcc<3nlum.— ^1  body  moves  aiont 
n  fixed  horizontal  axis  acted  on  by  gravity  (?vli/,  to  determine 
the  motion. 

Lot  ABO  be  a  section  of  the  body  made  by 
the  iilane  of  the  paper  passing  through  O, 
tlie  centre  of  gravity,  and  cutting  the  uxis 
of  rotation  iierpondioularly  at  0.  Lot  0  — 
the  angle  which  00  makes  with  the  vertical 
OY;  and  let  h  =  00,  t,  =  the  principiil 
radius  of  gyration,  and  M  =  the  mass  of 
the  body.  Then  hy  (4)  of  Art.  336,  we 
have 

20 


•iMfl-TT-- TlWiai 


''''"•""'  I'i  ii"r"'n'ri-  rn'  '  ■imrr 


468 


TH£  COirPOUND  PENDULVM. 


"~f 


Mgh  ain  d 


_       Mgh  Bin  6 


=  ~  ^;nnra  sin  e  [by  (2)  of  Art.  226],  (1) 

the  negative  sign  being  taken  because  9  is  a  decreasing 
function  of  the  time. 

This  etjuation  cannot  be  integrated  in  finite  terms,  but 
if  the  oscillations  be  small,  we  may  develop  sin  f>  and  reject 
all  powers  above  the  first,  and  (1)  will  become 


(Id 


gh 


*,»  +  A« 


e. 


(2) 


Multiplying  by  2  -^  and  integrating,  and  supposing  that 

the  body  began  to  move  when  6  was  equal  to  «,  (2) 
becomes 

rffl*  _      gh 

dP 


M(«»-n 


*,^  +  A« 


Hence  denoting  the  time  of  a  complete  oscillation  by  T, 
we  have 


/h"  +  k{' 


(8) 


which  gives  the  time  in  seconds,  when  h  and  Jfe,  are  meas- 
ured in  feet  and  g  =  32.18. 

When  a  heavy  body  vibrates  about  a  horizontal  axis,  by 
the  force  of  gravity,  it  is  called  a  compound  pendulum. 

CoE.  1. — If  we  suppose  the  whole  mass  of  the  compound 
pendulum  to  be  concentrated  into  a  single  point,  and  this 
point  connected  with  the  axis  by  a  medium  without  weight, 
it  becomes  a  simple  pendulum  (Art.  194).  Denoting  the 
distance  of  the  point  of  concentration  from  the  axis  by  I, 
wc  luvvc  for  the  time  of  an  oscillation,  by  (1)  of  Art.  194, 


afl 


of  Art.  326],  (1) 

is  a  decreasing 

inite  terms,  but 
sin  0  and  reject 
me 

(«) 

i  supposing  that 
equal  to  «,  (2) 


oscillation  by  T, 


(3) 

and  yfci  are  meas- 

orizontal  axis,  by 
d  pendulum. 

1  of  the  compound 
;le  point,  and  this 
m  without  weight, 
t).  Denoting  the 
•om  the  axis  by  I, 
y  (1)  of  Art.  194, 


■ '  '^fSISWI^SWiSSW' 


CBXTSES  OF  OSCILLATION  AND  SUSPENSION.      459 


"V^- 


If  the  point  be  so  chosen  that  the  simple  pendu- 


lum will  perform  an  oscillation  in  the  same  time  as  the 
comijound  pendulum,  these  two  expressions  for  the  time  of 
an  oscillation  must  be  equal  to  each  other,  and  wo  shuU 
liave 


1  = 


h 
A  +  ^  =  00', 


(4) 


(0'  being  the  point  of  concentration). 


Cob.  2. — This  length  is  called  the  length  of  the  simple 
equivalent  pendulum ;  the  point  0  is  called  the  centre  of 
suspension ;  the  point  0',  into  which  the  mass  of  the  com- 
pound pendulum  must  be  concentrated  so  that  it  will 
oscillate  in  the  same  time  as  before,  is  called  the  centre  of 
oscillation;  and  a  line  through  the  centre  of  oscillation 
and  ps-i-allel  to  the  axis  of  suspension  is  called  an  axis  of 
oscillation. 

From  (4)  we  have 

{l-7i)7i  =  *,»; 


or 


GO'.  GO  =  *,». 


(5) 


Now  (5)  would  not  be  altered  if  the  place  of  0  and  0' 
were  interchanged;*  hence  if  O'  be  made  the  centre  of 
suspension,  then  0  will  be  the  centre  of  oscillation.  Thus 
tlie  centres  of  oscillation  and  of  suspension  are  convertible, 
and  tfie  time  of  oscillation  about  each  is  the  same. 

CoR.  3. — Putting  the  derivative  of  I  with  respect  to  h  in 
(4)  equal  to  zero,  and  solving  for  h,  wo  got 

h  —  k^, 


460 


EXAMPLES. 


wliich  makes  I  a  minimum,  and  therefore  makes  /  a  mini- 
mum. Hence,  when  the  axis  of  suspension  jjasses  thmuijh 
the  principal  centre  of  gyration  the  time  of  oseillaiion  is  a 
minimum. 

Rem. — The  problem  of  dctnrmining  the  law  under  which  a  heavy 
body  swings  about  a  liorizontftl  nxia  is  one  of  tlie  most  iniiwrtnnt  in 
tlic  lilgtory  of  science.  A  simple  pendulum  is  a  thing  of  theory  ;  our 
accurate  Imowlodge  of  the  acceleration  of  gravity  depends  tberoforo 
on  our  understanding  the  rigid  or  compound  pendulum.  This  was 
th<>  first  problem  to  which  IVAlembert  applied  his  principle. 

The  problem  was  called  in  the  days  of  D'Alembert,  the  "  centre  of 
oscillation."  It  was  required  to  find  if  there  were  a  point  at  which 
the  whole  mass  of  the  body  might  be  concentrated,  so  as  to  form  u 
simple  pendulum  whose  law  of  ot^cillation  was  the  same. 

The  jxiRitlon  of  the  centre  of  oscillation  of  a  body  was  first  correctly 
determined  by  Huyghens  and  published  at  Wans  in  1678.  As 
D'Alembert's  principle  was  not  known  at  that  time,  Huyghens  had  to 
discover  some  principle  for  himself.* 


EXAMPLES. 

1.  A  material  straight  line  oscillates  about  an  axis  jier- 
pendicnlar  to  its  length ;  find  the  length  of  the  equivalent 
simple  pendulum. 

Let  'ia  =  the  length  of  the  line,  and  h  the  distance  of  its 
centre  of  gravity  from  the  point  of  suspension.    Then  since 


ij«  =  ^-,  we  have  from  (4) 


l  =  h  + 


U 


(1) 


Cor.  1. — If  the  point  of  suspension  be  at  the  extremity 
of  the  line  (1)  becomes 


*  Rontb's  Rigid  DTDiimlcB,  ]>,  69, 


EXAMPLES. 


461 


makes  t  a  mini- 
fy 2Jasses  through 
f  oscillation  is  a 


ndor  which  a  heavy 
!  most  iniiKirtnnt  in 
liing  of  tlieory  ;  our 
y  depends  therefore 
oduluiu.  This  was 
I  principle, 
ibert,  the  "  centre  of 
e  a  point  at  which 
ted,  BO  as  to  form  a 
9  same. 

dy  was  first  correctly 
tans  in  1678.  As 
le,  Huyghens  had  to 


xbout  an  axis  -pev- 
of  the  equivalent 

the  distance  of  ite 
ision.    Then  sine© 


(1) 

le  at  the  extremity 


that  is,  the  length  of  the  equivalent  simple  pendulum  is 
two-thirds  of  the  length  of  the  rod. 

Cob.  2.— Let  /t  —  |a ;  then  (1)  becomes 

I  =  !«.. 

Hence,  the  time  of  an  oscillation  is  the  same,  whether  tho 
line  bo  suspended  from  one  extremity,  or  from  a  point  one- 
third  of  its  length  from  the  extremity.  This  also  iilustnites 
the  convertibility  of  the  centres  of  oEoillation  and  of  sus- 
pension (See  Cor.  2). 

Cob.  3.— If  h  =  10a,  then  (1)  becomes 

1  = 


2.  A  circular  arc  oscillates  about  an  axis  through  its 
middle  point  i^erpendicular  to  the  plane  of  the  arc.  Prove 
that  the  length  of  the  simple  equivalent  pendulum  is 
independent  of  the  length  of  the  arc,  and  is  equal  to  twice 
the  radius. 

From  Ex.  2,  Art.  333,  we  have 


P 


^  /^       sin  <*\   o 


From  Ex.  1,  Art.  79,  we  have 


h  —  a  —  a 


sin  a 


Therefore  (4)  becomes 

«  ..  /^       sin  «\         /i       sin  «\       „ 
I  =  20" ^1 ^— J  -7-a\i ~j  =  2rt. 


,finn 


Iff 


LENOTII  OF  TBTS  SECONffS  PEXDVLCM. 


0jl 


m' 


Sll 


3.  A  right  cono  oscillates  about  an  axis  poasiug  through 
its  vertex  and  perpendicular  to  its  own  axis ;  it  is  required 
to  find  the  length  of  the  simple  equivalent  pendulum,  (1) 
when  h  is  the  altitude  of  the  cone  and  b  tlie  radius  of  the 
baflo,  and  (;j)  when  the  altitude  =  the  radius  of  the  base  =  h. 

Ans.  (1)  —gy-  ;  (3)  h. 

That  is,  in  t!ie  second  cono,  the  centre  of  oscillation  is  in 
the  centre  of  the  bawc ;  so  that  the  times  of  oscillation  are 
equal  for  axes  through  the  vertex  and  the  centre  of  the 
base  perpendicular  to  the  axis  of  the  cone. 

4.  A  sphere,  radius  a.  oscillates  about  an  axis  ;  find  the 
length  of  the  simple  eciuivalent  pendulum,  (1)  when  the 
axis  is  tangent  to  the  sphere,  (3)  when  it  is  distant  10« 
from  the  centre  of  the  sphere,  and  (3)  when  it  is  distant 

-  fi'om  the  centre  of  the  sphere.  , 

Ans.  {I)  la;  (2)  W«;  O^)  V«. 

238.    The   Length    of    the    Second's    Penduluir. 

Determined  Experimentally. — The  time  of  oscillulion 

k  * 
of  a  compound  pendulum  depends  on  h  +  -,-  by  (4)  of 

Art.  337.  But  there  are  difficulties  in  the  way  of  determin- 
ing /»  and  ky  The  centre,  G,  can  not  be  got  at,  and,  as 
every  body  is  more  or  less  irregular  and  v&riablo  in  density, 
h^  cannot  bo  calculated  with  sufficient  accuracy.  These 
(piantities  must  therefore  be  determined  from  experiments. 
Bessel  observed  the  times  of  oscillation  about  different 
axes,  the  distances  between  which  wore  very  accurately 
known.  Captain  Kator  employed  the  property  of  the 
convertibility  of  the  centres  of  suspension  and  oscillation 
(Art.  237,  Cor.  2),  as  follows : 

'  Let  the  pendnlum  consiHt  of  an  ordinary  stralglit  bar,  CO,  and  a 
small  weight,  m,  wliich  may  be  clamped  to  it  l>y  means  of  a  screw, 
uud  shifted  fruiu  ouu  positiuu  to  auuiher  cm  the  peuduluiu.     At  tho 


(2)  h. 


DVLOM. 

paasiug  through 
8 ;  it  is  required 
it  pendulum,  (1) 
Llie  radius  ai  the 
8  of  the  base  =  h. 

f  oscillation  ia  in 
of  oscillation  are 
he  centre  of  the 


m  axis  ;  find  the 

m,  (1)  when  the 

it  is  distant  lOrt 

vhen  it  is  distant 


W«; 


(:i) 


id's   Penduluir. 

nie  of  oscillation 

I  +  ^^  by  (4)  of 

way  of  detcrmin- 
be  got  at,  and,  as 
iriablo  in  density, 
iiccnraoy.  These 
Prom  experiments. 
I  about  different 
9  very  accurately 

property  of  the 
m  and  oscillation 

aigbt  bar,  CO,  and  a 
ly  means  of  a  screw, 
B  penduluiu.    At  llio 


"ipwn'  ."r^TO  tKV^^r^pKV 


LENGTH  OF  THE  SECOND'S  PENDULtm. 


403 


[Tin* 


o     I 


>ifl 


l1]»» 


Fls.94 


points  C  and  O  in  tvjo  triangular  aper- 
tures, at  the  distance  I  apart,  let  two  knife 
edges  of  hard  steel  bo  placed  parallel  to 
each  other,  and  at  right  angles  to  the 
pendulum,  so  that  it  mny  vibrate  on  either 
of  them,  OS  in  Fig.  04.  Let  m  be  shifted 
till  it  is  found  that  the  times  of  oscillatiuu 
about  0  and  O  are  exactly  the  same.  It 
remains  only  to  measure  CO,  and  observe 
the  time  of  oscHlation.  The  distance  be- 
tween the  two  points  C  and  O  is  the  length 

of  the  simple  equivalent  pendulum.  This  distance  between  the  knife 
edges  was  measured  by  Captain  Kater  with  the  greatest  care.  The 
mean  of  three  measurements  differed  by  less  than  a  ten-thoustindth 
of  an  inch  from  each  of  the  separate  measurements. 

The  time  of  a  single  vibration  cannot  be  observed  directly,  because 
this  would  require  the  fraction  of  a  second  of  time  as  shown  by  tho 
clock,  to  be  estimated  either  by  the  eye  or  ear.  The  difflouity  may 
be  overcome  by  observitg  the  time,  say  of  a  thousand  vibrations,  and 
thus  the  error  of  the  time  of  a  single  vibration  is  divided  by  a 
thousand.  The  labor  of  so  much  counting  may  however  bo  avoided 
by  the  use  of  "  the  method  of  coincidences."  The  pendulum  is  placed 
iu  f.-ont  of  a  clock  pcudulum  whob-  time  of  vibration  ia  slightly 
different.  Certain  marks  made  on  the  two  pendulums  are  observud 
by  a  telescope  at  the  lowest  point  of  their  arcs  of  vibratior  The  field 
of  view  is  limited  by  a  diaphragm  to  a  narrow  aperture  "cross  which 
the  marks  are  seen  to  pass.  At  each  succeeding  vibration  one 
pendulum  follows  the  other  more  closely,  and  at  Inst  its  mark  is 
completely  covered  by  the  other  during  their  passage  acrosr  the  field 
of  view  of  the  telescope.  After  a  few  vibrations  it  appears  again 
preceding  the  other.  In  the  interval  from  one  disappearance  to  the 
next,  ono  pendulum  has  made,  as  nearly  as  possible,  one  complete 
oscillation  more  than  the  other.  In  this  manner  530  half -vibrations  of 
a  clock  pendulum,  each  equal  to  a  second,  were  found  to  correspond  to 
533  of  Captain  Eater's  pendulum.  The  ratio  of  the  times  of  vibra- 
tion of  the  pendulum  and  the  clock  pendulum  may  thus  be  calculated 
with  extreme  accuracy.  The  rate  of  going  of  the  clock  must  then  bo 
found  by  astronomical  means. 

The  time  of  vibration  thus  found  will  require  several  corrections 
which  are  called  "reductions."  For  instance,  if  the  oscillation  be 
not  so  small  that  we  can  put  sind  —  0  in  Art.  237,  wo  must  make  a 
reduction  to  infinitely  small  arcs.    Another  reduction  is  necessary  if 


fill 


m 


r<,l 


M' 


j%t*«3»iri<os»iii»>oi«maa4K»S;»iK<»»a^ 


JaSJ)^ 


4G4       MOTION  OF  A   BODY  WHEN   UNCO NSTBAI NED. 


IIP 


we  wish  to  reduce  the  rpsult  to  what  It  would  liAve  been  at  the  level 
of  the  sea.  Tha  altfaction  uf  the  interveDing  land  Toay  be  allowed 
for  by  Dr.  Young's  rule,  (Phil.  Trans.,  1819).  We  laay  thuu  obtain 
the  force  of  gravity  at  the  level  of  the  sea,  annpoeing  all  the  laud 
above  this  level  were  cut  ott'  and  the  sea  cuaBtraincd  to  keep  its 
present  level.  As  the  level  of  the  sea  is  altered  by  the  attraction  of 
the  land,  further  correcticas  au)  still  necessary  if  we  wish  to  reduce 
the  result  to  the  snrfaco  of  that  spheroid  which  most  nearly  repre- 
sents the  earth.  See  Routh's  lligid  Uynamios,  p.  77.  For  the  details 
of  this  experiment  the  Htudent  \a  referred  to  the  Plill.  Trans,  for  1818, 
and  to  Vol.  X. 

<    239.  Modon  of  a  Body  when  Unconstrained.— If 

an  impulf'o  be  commuuicated  to  any  point  of  a  freo  body 
ill  a  direction  not  pfwsing  through  the  centre  of  gravity,  it 
will  ])roditce  both  translation  and  rotation. 

Let  P  be  the  impulse  imparted  to 
the  body  at  A.  At  B,  ou  the  opposite 
eide  of  the  centre  G,  a  distance  GB  ^ 
r=  AG,  let  two  opposite  impulses  be  >  j 
applied,  each  equal  to  \P ;  they  will 
not  alter  the  effect.  Now  if  \P 
applied  at  A  is  combined  with  the  ^7' 
at  B  which  acts  in  the  same  diraotion,  their  resultant  is  P, 
acting  at  G  and  in  the  «.ame  direction,  and  this  produces 
translation  only.  The  remaining  ^P  at  A  combined  with 
the  remaining  \P  at  B,  which  acts  in  the  opposite  direc- 
tion, form  a  couple  wuich  produces  rotation  about  the 
centre  G. 

Hence,  when  a  body  receives  an  impulse  in  a  'lirection 
which  doer  not  pnifS  through  the  centre  of  gravity,  thxt  centre 
ml'  assume  n  motton  of  trnndalion  as  ihongh  the  ImpuUe 
were  applied  immediately  to  it ;  and  the  body  will  have  a 
motion  of  rotation  about  the  centre  of  gravity,  as  though 
Uiat  jwint  were  fixed. 

240.  Centre  of  Fercusaion.  -Axis  of  Spontaneous 
Rotation. — Lot  Jli>  rcprcsuut  the  impulse  imprensod  upon 


IP 
FI0.9S 


BAILED. 

3een  at  the  level 
I  tnay  be  allowed 
!uay  thiu  obtain 
sing  all  the  laud 
aiued  to  keep  Uh 
-  the  attraction  of 
we  wisb  to  reduce 
noBt  nearly  repre- 
,  For  the  dutoila 
I.  Trana.  for  1818, 


aBtrained.— II 

of  a  free  body 
re  of  gravity,  it 


rihjmmmm'mm^  ■ 


'fm 


If 

Ftg.9S 

rc&ultant  is  P, 
id  this  produces 
L  combined  with 
I  opposite  direfi- 
xion  about  the 

J  tu  a  'lirecHon 
ivity,  th  tt  centre 
ugh  the  mptihe 
'jody  will  have  a 
vily,  as  though 

f  BpontaneouB 

imprt'ttsiod  ui>ou 


CENTRE  OP  PERCUSSION. 


4G5 


the  body  (Fig.  95)  whose  mass  is  M,  and 
h  the  perpendicular  distance,  00,  from 
the  centre  of  gravity,  G,  to  the  line  of 
action,  OP,  of  the  impnlse.  The  centre 
of  gravity  will  assume  a  motion  of  trans- 
lation with  the  velocity  v,  in  a  direction 
parallel  to  that  of  the  impulsive  force. 
Then  from  (3)  of  Art  236,  we  havo  for  the  angular 
velocity 

Mvh  _  vh 


Fl0:M 


w  = 


The  absolute  velocity  of  each  point  of  the  body  will  bo 
compounded  of  the  two  velocitieft  of  translfition  and  rota- 
tion. TL  "  point  0,  for  example,  to  which  the  impulse  is 
applied,  haa  a  velocity  of  translation,  Oa,  equaUto  that  of 
the  cen*^^re  of  gravity,  and  a  velocity  of  rotation,  ab,  about 
the  centre  of  gravity ;  so  that  the  velocity  of  any  point  at 
tf  distance  a  from  the  centre,  0,  will  bo  expressed  by 
I?  i:  aw ;  the  upper  x  -  lower  sign  being  taken  according  as 
the  point  is,  or  is  uot,  on  the  same  side  of  the  centra  of 
gravity  as  the  point  0.  Thus,  if  we  consider  the  motion  of 
the  body  for  a  vevy  short  interval  of  time,  the  line  000 
will  assume  the  position  bOC,  the  point  C  remaining  at 
rest  during  this  interval ;  that  is,  while  the  point  0  would 
ho  carried  forward  over  the  line  Cc  by  the  motion  of  trans- 
lation, it  would  bo  carried  backward  through  the  same 
distance  by  the  motion  of  rotation.  Ueuce,  since  the  abso- 
lute velocity  of  O  is  zero,  wo  have 


ow  .-=  0 ; 


«  =  -  =  ^  ; 
w       A  ' 


(1) 


lud  hence  denoting  00  Ity  I  we  havo 


^uM# 


466 


AXIS  OF  SPOyTAffXOUS  fiOTATIOff. 


1  = 


(3) 


Now  if  there  had  been  a  fiied  axis  throngh  C  perpen- 
dicular to  the  plane  of  motion^  the  initial  motion  would 
liave  been  precisely  the  same,  and  this  fixed  axis  evidently 
would  p.ot  have  received  any  pressure  from  the  impulse. 

"When  a  rigid  body  rotates  about  a  fixed  axis,  and  the 
bod^  can  be  so  struck  that  there  is  no  pressure  on  the  axis, 
any  point  in  the  line  of  action  of  the  force  is  called  a  centre 
of  percussion. 

When  the  line  of  action  of  the  blow  is  given  a^  d  the 
body  is  free  from  all  constraint,  so  that  it  is  capable  of 
translation  as  well  as  of  rotation,  the  axis  about  which  the 
body  begins  to  turn  is  called  the  axis  of  spontamous  rota- 
tion. It  obviously  coinoidcs  with  the  position  of  the  fixed 
axis  iu  tbtv  first  case. 

Cou.  1. — From  (1)  we  have  * 

ah  =  GC'OQ  =  *,»; 

hence  the  points  0  and  C  are  convertible,  that  is,  if  the 
(ms  of  rotation  be  supposed  to  pass  throvgh  the  point  0, 
the  centre  of  spontaneous  rotation  will  coincide  with  the  sen- 
tre  of  percussion. 

Cob.  3, — From  (2)  it  follows,  by  comparison  with  (4)  of 
Art.  237,  that  if  the  axis  of  spontaneous  rotation  coincides 
wi*h  the  aids  of  suspension,  the  centre  of  percussion  coin- 
cides with  the  centre  of  oscillation. 

Son. — It  is  evident  that  if  there  be  a  fixed  obstacle  at  0, 
and  '<-  be  struck  by  the  body  OC  rotating  about  a  fixed 
axis  through  C,  the  obstacle  will  recei"e  the  whole  force 
of  the  moving  body,  and  the  axis  will  not  receive  any. 
Hence  the  centre  of  percuss Icn  also  dct?rrainos  the  position 
'  in  which  a  fixed  obBtuclo  niunt  bo  ])laccd,  on  which  if  the 
rotating  body  impinges  nud  iu  brought  to  rest,  the  axis  of 
rotation  will  suflior  no  pretwure. 


TIOH. 

(3) 

rough  C  perpcn- 
ial  motion  would 
jd  axis  evidently 
1  the  impulse, 
ced  axis,  and  the 
jsure  on  the  axis, 
I  is  called  a  centre 

is  given  a'  d  the 
it  it  is  capable  of 

about  which  the 
spontamous  rota- 
litiou  of  the  lixed 


EXAMPLES. 


467 


lie,  that  is,  if  the 
ovgh  the  point  O, 
icide  with  the  cen- 

arison  with  (4)  of 

rotation  coincides 

\f  percussion  coin- 

ixed  obstacle  at  0, 
,ing  about  a  fixed 

0  the  whole  force 

1  not  receive  any. 
mines  the  position 
1,  on  which  if  the 
,0  rest,  the  axis  of 


An  axis  through  the  centre  of  gravity,  parallel  to  the 
axis  of  spontaneous  rotation,  is  called  the  axis  of  instantane- 
ous rotation.    A  free  body  rotateo  about  this  axis  (Art,  239). 

* 

EXAMPLF.  S. 

1.  Find  the  centre  of  percussion  of  a  circular  plate  of 
radius  a  capable  of  rotating  about  an  axis  which  touches  it. 

Here  /;,"  =  j,  and  h  =  a.     Hence  from  (3)  we  have 
I  =  a  +  ■T  =  {a. 

2.  A  cylinder  is  capable  of  rotating  about  the  diameter 
of  one  of  its  circular  ends ;  find  the  centre  of  percussion. 
Let  a  =  its  length,  a.nd  b  =  the  raf'ius  of  ltd  base. 

3i»  +  4a^ 


Ans.  I  =z 


Ga 


Henca  if  2&'  =  2a\  the  centre  of  iwrcussion  wiE  be  at 
the  end  of  the  cylinder.  If  b  is  very  small  compared  with 
a,  I  =  ^i;  thus  if  a  straight  rod  of  small  transverse  section 
is  held  by  one  end  in  the  hand,  I  gives  the  point  at  which 
it  may  be  struck  so  that  the  hand  will  receive  no  jar. 

2^1.  The  Principfil  Radius  of  Gyratior  Deter- 
mined Practioally. — Mount  the  body  upon  an  axis  not 
passing  through  the  centre  of  gravity,  and  cause  it  to 
oscillate  ;  from  the  number  of  oscillations  performed  in  a 
given  time,  say  an  hour,  the  time  of  one  oscillation  ia 
known.  Then  to  find  h,  which  is  the  distance  from  the 
axis  to  the  centre  of  gravity,  attach  a  spring  balance  to  the 
lower  onil,  and  bring  the  centre  of  gravity  to  a  horizontal 
])lane  through  the  axis,  which  position  will  be  indicated  by 
the  maximum  reading  of  (he  balance.  Knowing  the  maxi 
mum  reading,  R,  of  the  balance,  the  weight,  H",  of  the 
body,  and  the  distance,  a,  from  the  axis  of  snKiHjusion  to 


IP" 
i 


468 


TUB  BALLISTIC  PENDTTLVM. 


tho  point  of  attachment,  we  have  from  the  principle  of 
moments,  Ra  =  Wh,  from  which  h  is  found.  Substitut- 
ing in  (3)  of  Art  237,  this  value  of  h,  and  for  T  the  time 
of  an  oscillation,  kt  becomes  known. 

242.  The  Ballistic  Fendnlnm. — An  interesting  aj)- 
plication  of  the  principles  of  the  compound  pendulum  is 
the  old  way  of  determining  the  velocity  of  a  bullet  or  can- 
non-ball. It  is  a  matter  of  considerable  importance  in  tho 
Theory  of  Gunnery  to  determine  the  velocity  of  a  bulk '  as 
it  issues  from  the  mouth  of  a  gun.  It  was  to  determine 
this  initial  velocity  that  Mr.  Robins  about  1743  invented 
the  Ballistic  Pendulum.  This  consists  of  a  large  thick 
heavy  mass  of  wood,  suspended  from  a  horizontal  axis  in 
tiie  shape  of  a  knife-edge,  after  the  manner  of  a  compound 
pendulum.  The  gun  is  so  placed  that  a  ball  projected 
from  it  horizontally  strikes  this  pendulum  at  rest  at  a  cer- 
tain point,  and  gives  it  a  certain  angular  velocity  about  its 
axis.  The  velocity  of  the  ball  is  itself  too  great  to  be 
measured  directly,  but  the  angular  velocity  communicated 
to  tho  pendulum  may  be  made  as  small  as  we  please  by 
increasing  its  bulk.  The  arc  of  oscillation  being  meas- 
ured, the  velocity  of  the  bullet  can  be  found  by  calcu- 
lation. 

Tho  time,  which  the  bullet  takes  to  penetrate,  is  so  short 
that  we  may  suppose  it  completed  before  the  pendulum  has 
sensibly  moved  from  its  initial  position. 

Let  M  be  the  riass  of  the  })eudulum  and  ball ;  m 
that  of  tho  ball ;  v  tho  velocity  of  tlie  ball  at  the  instant  of 
impact ;  h  the  distance  of  the  centre  of  gravity  of  tiio  pen- 
dulum and  ball  from  the  axis  of  suspension  ;  a  the  distance 
of  tho  point  of  impact  from  the  axis  of  susfwusion  ;  w  tho 
angular  vi'octity  due  to  tho  blow  of  the  ball,  and  k  the 
radius  of  gyration  of  the  pendulum  and  ball.  Then  since 
t  h(>  initial  velocity  of  tiio  bullet  is  v,  its  impulse  is  nieaaurod 
l.v  iiiv,  au(i  therefore  from  (3)  of  Art.  ^30  wu  have  for  the 


TUB  UAhLlSTIC  PEXDVhUM. 


469 


be  principle  of 
iid.  Substitut- 
for  T  the  time 


interesting  ajv 
id  pendulum  is 
I  bullet  or  can- 
jortance  in  the 
y  of  a  bulk '  as 
s  to  deterniino 

1743  invented 
I  a  large  thick 
rizontal  axis  in 
of  a  compound 

ball  projected 
t  rest  at  a  cer- 
locity  about  its 
too  great  to  be 

communicated 
as  we  please  by 
on  being  meas- 
bund  by  calcii- 

rate,  is  so  short 
3  pendulum  has 

L  and  ball;  m 
it  the  instant  of 
niy  of  the  pon- 
;  a  the  distance 
pension  ;  w  the 
l)all,  and  k  the 
,11,  Then  since 
iilse  is  meuaurcd 
ve  have  for  the 


initial  angular  velocity  generated  in  the  pendulum  by  this 
impulse, 

mva 


6)  = 


M-a' 


(1) 


and  from   (I)  of  Art.  237  we  have  for  the  subsequent 
motion 


0^  qh    .    ^ 


(8) 


Integrating,  and  observing  that,  if  «  bo  the  angle  through 
which  the  iiendulum  moves,  we  have  ^  =  w  when  ©  =  0, 

and  ^  =  0  when  0  =  «,  (2)  becomes 


cos  a). 


Eliminating  w  between  (1)  and  (3)  we  have 

%Mh    r-r    .    a 

V  = Vffh  sm  - , 

ma      "  2 


(3) 


(4) 


from  which  v  becomes  known,  since  all  the  quantities  in 
the  second  member  may  be  observed,  or  are  known. 

Wo  may  determine  a  as  follows :  At  a  point  in  the  jwn- 
dulum  at  a  distance  h  from  the  axis  of  suspension,  attach 
the  end  of  a  tape,  and  let  the  rest  of  the  tape  be  wound 
tightly  round  a  reel ;  as  the  pendulnm  ascends,  let  a  length 
c  be  unwound  from  the  reel ;  then  c  is  the  chord  of  tlio 

angle  «  to  the  radius  h,  so  that  c  ~  2h  sin  |,  which  in  (4) 
givce 

Mkc     fa 

The  values  of  k  and  //  may  bo  dot«rmmed  as  in  Art.  241. 
If  the  mouth  of  the  gun  is  placed  near  to  the  pendulum. 


470 


ROTATION  OF  A  HEAVY  BODY. 


m 


the  value  of  v,  given  by  (5),  must  be  nearly  the  velocity  of 
projection. 

The  velocity  may  also  be  determined  in  the  following 
manner :  Let  the  gun  be  attached  to  a  heavy  pendulum ; 
when  the  gun  is  discharged  the  recoil  causes  the  pendulum 
to  turn  round  its  axis  and  to  oscillate  through  an  arc 
which  can  be  measured  ;  and  ihe  velocity  of  the  bullet  can 
be  deduced  from  the  magnitude  of  this  arc.  (See  Price's 
Anal.  Mech's,  Vol.  II,  p.  231.) 

Before  the  invention  of  the  balliatic  pendulnm  by  Mr.  Robins  in 
174.'},  but  little  progress  bad  been  made  in  the  true  theory  of  mLitary 
projectiles.  Robins'  New  Principles  of  Gunnery  was  soon  translated 
into  several  languages,  and  Euler  added  to  his  translation  of  it  into 
German  an  extensive  commentary  ;  the  work  of  Baler's  being  again 
tranalatSd  into  English  in  1784.  The  experiments  of  Robins  were 
all  conducted  with  musket  balls  of  about  an  ounce  weight,  but  they 
wore  afterwards  continued  during  several  years  by  Dr.  Button,  who 
used  cannon-balls  of  from  one  to  nearly  three  pounds  in  weight. 
Ilutton  used  to  suspend  his  cannon  as  a  pendulum,  and  measure  the 
angle  through  which  it  was  raised  by  the  discliarge.  His  experi- 
monts  ar<)  still  regarded  as  some  of  the  most  trustworthy  on  smooth- 
bore guns.  See  Routh's  Rigid  Dynamics,  p.  04,  also  Encyclop»dia 
Britannica,  Art.  Gunnery. 

243.  Motion  of  a  Heavy  Body  about  a  Horizon- 
tal Axle  through  its  Centre. — Let  the  body  be  a  spliere 
whose  radius  is  R,  and  weight  W,  and  \z'c  a  weight  P  be 
attaclied  to  a  cord  wound  round  the  circumference  of  a 
wheel  on  ^ho  same  axle,  the  radius  of  the  wheel  being  r ; 
rofiuircd  the  dif;taiice  ptissed  over  by  P  in  t  seconds. 

From  (4)  of  Art.  230  wo  have 

cPd  _         Prg 
d^  ~  Wkf+  Pr^' 


I     Multiplying  by  dt  and  integrating  twice,  we  have 

Prgt^ 


$  = 


i{Wk*  +  Pr^y 


(1) 


m 


>r. 

y  the  velocity  of 

n  the  following 
!avy  pendulum ; 
cs  the  pendulum 
through  an  arc 
ii  the  bullet  can 
re.    (See  Price's 

1  by  Mr.  Robi-s  in 
e  theory  of  mLitary 
eras  soon  translated 
ranslation  of  it  into 
Euler's  being  again 
its  of  Robins  were 
ce  weight,  but  they 
by  Dr.  Hutton,  who 
I  pounds  in  weight, 
m,  and  measure  the 
charge.  His  experi- 
stworthy  on  smooth- 
;,  also  Encyclopsedia 

)OQt  a  Horizon- 

e  body  be  a  sphere 
1;  a  -weight  P  be 
ircumfercnoe  of  a 
le  wheel  being  r ; 
t  seconds. 


ii|gB;i^<lW»Vlwy>jMI|>!«j!MMW 


!e,  we  have 


(1) 


FXAMPLBS. 


471 


the  constants  being  zero  in  both  integrations,  since  the  body 
starts  from  rest  when  t  =  0.    The  space  will  be  rd. 

EXAMPLES. 

1.  Let  the  body  be  a  sphere  whose  radius  is  3  ft.  and 
weight  500  lbs.;  let  P  be  50  Ibe.,  and  the  radius  of  the 
wheel  6  ins.;  required  the  time  in  which  the  weight  P  will 
descend  through  50  ft.    (Take  g  =  32.) 

Ans.  21  seconds. 

2.  Let  the  body  be  a  sphere  Whose  radius  is  14  ins.  and 
weight  800  lbs.;  let  it  be  moved  by  a  weight  of  200  lbs. 
attached  to  a  cord  wound  round  a  wheel  the  radius  of 
which  is  one  foot ;  find  the  number  of  revolutions  of  the 
sphere  in  eight  seconds.    (Take  g  =  32.)         Ans.  51.3. 

244.  Motion  of  a  Wheel  and 
Azle  when  a  Given  Weight  P 
RaiseH   a  Given  Weight   W.— Let 

the  weights  P  and  W  be  atfached  to 
cords  wound  round  the  wheel  and  axle, 
respectively,  (Fig.  97)  ;  let  R  and  r  be 
be  the  radii  of  the  wheel  and  axle,  and 
w  and  w'  their  weights;  required  the 
angulai'  distance  passed  over  in  t 
seconds. 
From  (4)  of  Art.  236,  we  have 


I 

w 


Fig.S7 


PR-Wr 


PR'  +  Wt^  +  itoR^  +  \w'r^ 


</; 


{PR  -  Wr)  t' 


Pm-^Wr'*  +  \wR?  +  i«)V 


y- 


(1) 


(2) 


EXAMPLE. 


Let  the  weight  P  =  30  lbs.,  If  =  80  lbs.,  w  =  8  lbs., 
and  w'  =  4  lbs.;  and  let  R  and  r  be  10  ins.  and  4  ins.; 


472 


MOTION  AUOXTT  A    VERTICAL  AXIS. 


required  (1)  the  space  passed  over  by  P  in  12  seconds  if  it 
starts  from  rest,  and  (2)  the  tonsiona  Tand  T'  of  the  cords, 
supporting  P  and  W.    (Take  g  =  32.) 
Ann.  (1)  97.79  ft.;  (2)  T  :=  31.28  lbs.;  T'  =  78.64  Iba. 


T 


\ 


-e- 


Fig.SS 


245.  Motion  of  a  Rigid  Body 
about  a  Vertical  Aada— Let  AB 
be  a  vertical  axis  about  which  the 
body  0,  on  the  horizontal  arm  ED, 
revolves,  under  the  action  of  a  con- 
stant horizontal  force  F,  applied  at 
the  extremity  E,  perpendicular  to 
ED.  T^t  M  be  the  mass  of  the  body  whose  centre  is  C, 
and  r  and  it  i^he  distances  ED  and  CD,  respectively.  Then 
from  (4)  of  Art.  ii'i".  we  have 

dp  -  jn  (k,» +  w\' : 

ft 

Multiplying  by  di  and  integrating  twice,  obsorviaj  that 
the  constants  of  both  integrations  are  zero,  we  have   "     ' 


d  = 


FrP 


which  is  the  angular  space  passed  over  in  t  seconds. 


(1) 


EXAMPLE. 

Let  the  body  be  a  sphere  whose  radius  is  2  ft,  whose 
weight  is  600  lbs.,  and  the  distance  of  whose  centre  from 
the  axis  is  8  ft.,  and  let  i'  be  a  foree  of  40  lbs.  acting  at  the 
end  of  an  arm  10  ft  long;  find  (1)  the  number  of  revolu- 
t\on8  which  the  body  will  make  about  the  axis  in  10 
minutes,  and  (2)  the  time  of  one  revolution.  (Take 
9  -  32.)  Am.  (1)  9316.3  ;  (2)  6.2  sees. 


xrs. 


L3  seconds  if  it 
T'  of  the  cords, 

"  =  78.64  lbs. 


e 


Fis.M 

ose  centre  is  C, 
ectively.    Then 


5,  observiK  J  that 
wo  have 


<1) 


seconds. 


3  is  2  ft.,  whose 
iiose  centre  from 
lbs.  acting  at  the 
imber  of  revolu- 

the  axis  in  10 
olation.      (Take 

;  (3)  6.2  sees. 


Fifl.99 


BOOr  ROLLTNO  DOWN  AN  INCLTNBD  PLANE.       473 

246.  Body  Rolling  down  an  Inclined  Plane.— ^ 

homogeneous  sphere  rolls  directly  down  a  rough  inclined 
plane  under  the  action  of  gravity.    Find  the  motion. 

Let  Fig.  99  represent  a  section 
Oi  Ae  sphere  and  plane  made  by  a 
vertical  plane  passing  through  0, 
the  centre  of  the  sphere.  Let  «  be 
the  inclination  of  the  plane  to  the 
horizon,  a  the  radius  of  the  sphere, 
0  the  point  of  the  plane  which 
was  initially  touched  by  the  sphere 
at  the  point  A,  P  the  point  of  contact  at  the  time  /, 
ACP  =  0,  which  is  the  angle  turned  through  by  tho 
sphere,  m  =  tho  mass  of  the  sphere,  F  =  the  friction 
acting  upwards,  R  =  the  pressure  of  the  sphere  on  the 
plane.  Then  it  is  convenient  to  choose  0  for  origin  and 
OB  for  the  axis  of  x ;  hence  OP  =  x. 

The  forces  which  act  on  the  sphere  are  (1)  the  reaction, 
B,  perpeu'licular  to  OB  at  P,  (2)  the  friction,  F,  acting  at 
P  along  PO,  and  (3)  its  weight,  mg,  acting  vertically  at  0 
■^bo  centre.  Now  0  evidently  moves  along  a  straight  line 
paralk'  tq  the  plane ;  so  that  for  its  motion  of  translation 
we  have,  by  ic=plving  along  the  plane, 


a*X 

in^Ts  —  '^P  sin  «  —  F. 

d^         '  ' 


(1) 


The  sphere  evidently  rotates  about  itK  point  of  contact 
with  the  plane;  but  it  may  be  considered  as  Tot^ting  at 
any  instant  about  its  centre  G  as  fixed ;  and  the  miliar 
velocity  of  0  at  that  instant  in  reference  to  P  is  the  same 
as  that  of  P  in  reference  to  C.  From  (4)  of  Art.  236,  we 
have  for  the  motion  of  rotation 


»"V^  =  i^ 


(8) 


.^"smm^^mfiijitsmim-x^iimsif;: 


:im.a-x-!<iiiefw^sr'msmim'  t 


i.C 


i: 


474       BODr  ROLLTNO  DOWN  AN  INCLINED  FLANS. 

and  since  the  plane  is  perfectly  rough,  so  that  the  sphere 
does  not  slide,  we  have 

X  =  ofl.  (3) 


Multiplying  (1)  by  a  and  adding  the  result  to  (2),  we  get 

(4) 


ma^  +  mki*  ^  =  mag  sin  a. 


Differentiating    (3)   twice  we    get  ^^ 


united  to  (4)  gives 


rj2 


cPx 

Since  the  sphere  is  homogeneous,  A?,» 
becomes 

dh;       . 
^=^^Bin« 


d»d      ,  ,  , 
a^,   which 


(6) 

f«a,  and  (5) 

(6) 


which  gives  the  acceleration  down  the  plane. 

If  the  sphere  had  been  sliding  down  a  smooth  plane,  the 
acceleration  would  have  been  g  sin  «  (Art.  144) ;  so  that 
two-sevenths  of  gravity  is  used  in  turning  the  sphere,  and 
five-sevenths  in  urging  the  sphere  down  the  plane. 

Integrating  (6)  twice,  and  supposing  the  sphere  to  start 
from  rest,  we  have 

X  ~  ^g  •  sin  «5  •  <' 

which  gives  the  space  passed  over  in  the  time  t. 
Resolving  perpendicular  to  the  phne,  we  have 

R  =.  mg  sin  a. 

Cor. — If  the  rolling  body  were  a  circular  cylinder  with 
<  its  axis  horizontal,  then  k^^  =  \a*,  and  (.5)  becomes 


(7) 


) 


W  PLANE. 

0  that  the  sphere 

(3) 
ult  to  (3),  we  get 

(4) 


Q  a. 


a  j^^,  which 


(5) 
»  =  |<?a,  and  (5) 


(6) 


smooth  plane,  the 
Irt.  144) ;  so  that 
ig  the  sphere,  and 
he  plane, 
he  sphere  to  start 


me  t. 
Bre  have 


ular  cylinder  with 
(5)  becomes 

(7) 


IMPULSIVE  FORCE. 


476 


BO  that  one-third  of  gravity  is  used  in  turning  the  cylinder, 
and  two-thirds  in  urging  it  down  the  piano. 
From  (7)  we  have 


a;  =  ^  sin  a  •  /' 


which  gives  the  space  passed  over  in  the  time  t  from  rest. 


(8) 


247.  Motion  of  a  Falling  Body  under  the  Action 
of  an  Impulsive  Force  not  Directly  through  its 
Centre. — A  string  is  wound  round  the  circumference  of  a 
reel,  and  the  free  end  is  attached  to  a  fixed  point.  The  reel 
is  then  lifted  w/>  and  let  fall  so  that  at  the  moment  when 
the  string  becomes  tight  it  is  vertical,  and  tangent  to  tJie  reel. 
The  whole  motion  being  supposed  to  take  place  in  one  plane, 
determine  the  effect  of  th^  impulse. 

The  reel  at  first  will  fall  vertically  without  rotation. 
Let  V  be  the  velocity  of  the  centre  at  the  momer^t  when  the 
string  becomes  tight ;  f ',  «  the  velocity  of  the  centre  and 
the  angular  velocity  just  after  the  impulse ;  T  the  impul- 
sive tension ;  m  the  mass  of  the  reel,  and  a  its  radius. 

Just  after  the  impact  the  part  of  the  reel  in  contact  with 
the  string  has  no  velocity,  and  at  this  instant  the  reel 
rotates  about  this  part;  but  it  may  be  considered  as 
rotating  about  its  axis  as  fixed,  and  the  angular  velocity  of 
its  axis,  at  this  instant,  in  reference  to  the  part  in  contact 
is  the  same  as  that  of  the  latter  in  reference  to  the  former. 
The  impulsive  tension  is 


T  —  m  {v  —  v'). 


(1) 


Hence  from  (4)  of  Art  236,  we  have  for  the  motion  of 
rotation . 


myfcj'w  =z  m{v  —  v'). 


(8) 


»Mmi 


lUtfian 


■  i/i-'>-:l.<,if^^-^-/r\^-,',.:' 


478 


IMPULSIVE  FORCE. 


u 

lit' 


Sinco  the  part  of  the  rod  in  contact  with  the  string  has 
no  velocity  at  the  instant  of  impact,  we  have 


V  =  aw. 
Solving  (3)  and  (3)  we  have 


6)  =: 


av 


a>  +  *,» 


(3) 


(4) 


«.a 


If  the  reel  bo  a  homogeneous  cylinder,  ft^'  =  -,  and  we 

« 


have  from  (3)  and  (4) 


w  =  f-,    v' =  Iv,  (5) 

I* 


and  from  (1)  we  have  for  the  impulsive  tension, 

T  =  ^mv. 

Cob. — To  find  the  subsequent  motion.  The  centre  of  the 
reel  begins  to  descend  vertically ;  and  as  there  is  no  hori- 
zontal force  on  it,  it  will  continue  to  descend  in  a  vertical 
straight  line,  and  throughout  all  its  subsequent  motion  ?;ho 
string  will  be  vertical.  The  motion  may  therefore  be 
easily  investigated,  as  in  Art  246,  since  it  is  similar  to  the 
case  of  a  body  rolling  down  an  inclined  plane  which  is 
vertical,  the  tension  of  the  string  taking  the  place  of  the 

friction  along  the  plane.  Hence  putting  «  =  S'  '^^^ 
letting  the  friction  F  =  the  finite  tension  of  the  string,  we 
have,  from  (1)  and  (7)  of  Art.  346, 

■   '  F=  \mg,    and    ^  =  fer ; 


that  is,  the  finite  tension  of  the  string  is  one-third  of  the 


jjggg^^jIMi 


li  the  Btring  has 
(8) 


(4) 


,»  =  -,  and  we 


(6) 


lion. 


he  centre  of  the 
bere  is  no  hori- 
ind  in  a  vertical 
aent  motion  ^ho 
ay  therefore  be 
is  similar  to  the 
[  plane  which  is 
the  place  of  the 

ig  «  =  I'  ^^^ 
of  the  sti-ing,  we 


one-third  of  the 


EXAMPLES. 


477 


weight,  and  the  reel  descends  with  a  nniform  acceleration 

Since  the  initial  velocity  of  the  reel  from  (5)  is  \v,  we 
have,  for  the  space  descended  in  the  time  t  after  the  impact, 
from  (8)  of  Art  ''46, 

\v  +  \gfl.    (See  Kouth's  Rigid  Dynamics,  p.  131.) 


X 


EXA.MPL.BS. 

1.  A  thin  rod  of  steel  10  ft  long,  oscillates  about  an  axis 
passing  through  one  end  of  it ;  find  (1)  the  time  of  an 
oscillation,  and  (3)  the  number  of  oscillations  it  makes  in  a 
day.  Ans.  (1)  1.434  sec. ;  (3)  60254. 

2.  A  pc-ndulum  oscillates  about  an  axis  passing  through 
its  end ;  it  consists  of  a  steel  rod  60  ins.  long,  with  a  rect- 
angular section  i  by  J  of  an  inch ;  on  this  rod  is  a  steel 
cylinder  2  in.  in  diameter  and  4  in.  long;  when  the  ends  of 
the  rod  and  cylinder  are  set  square,  find  the  time  of  an 
oscillation.  Am.  1.174  sees. 

3.  Determine  the  radius  of  gyration  with  reference  to 
the  axis  of  suspension  of  a  body  that  makes  73  oscillations 
in  2  minutes,  the  distance  of  the  centre  of  gravity  from  the 
axis  being  3  ft  2  in.  Ans.  5.267  ft. 

4.  Determine  the  distance  between  the  centres  of  suspen- 
sion and  oscillation  of  a  body  that  oscillates  in  2J^  sec. 

Ans.  20.264  ft 

5.  A  thin  circular  plate  oscillates  about  an  axis  jfassing 
through  the  circumference  ;  find  the  length  of  the  simple 
equivalent  pendulum,  (1)  when  the  axis  touches  the  circl? 
and  is  in  its  plane,  and  (2)  when  it  is  at  right  angles  to 
the  plane  of  the  circle.  Ans.  (1)  Jw  ;  (2)  |«. 

6.  A  cube  oscillates  about  one  of  its  edges;  find  the 
length  of  the  simple  equivalent  pendulum,  the  edge  being 

=  ^«-  Ans.  |«  ^2. 


m 


:t3i!i"' 


liil 


■■m 

m 


■■W'[ 


478 


EXAMPLES. 


7.  A  prism,  whose  cross  section  is  a  square,  each  sido 
being  =  2n,  aud  whose  length  is  I,  oscillates  about  one  of 
its  upper  edges;  find  the  length  of  the  simple  equivalent 
pendulum.  Ans.  f  V'to*  +  P. 

8.  An  elliptic  lamina  is  such  that  when  it  swings  about 
one  latus  rectum  as  a  horizontal  axis,  the  other  Intus 
rectum  passes  through  the  centre  of  oscillation ;  prove, 
that  the  eccentricity  is  ^. 

9.  The  density  of  a  rod  varies  as  the  distaiice  from  one 
end  j  find  the  axis  perpendicular  to  it  about  which  the 
time  of  oscillation  is  a  minimum,  I  being  the  length  of  the 
rod. 

Ans.  The  distauce  of  the  axis  from  the  centre  of  gravity 

is  \  Va. 

0 

10.  Find  the  axia  about  which  an  elliptic  lamina  must 
oscillate  thai  the  time  of  oscillation  may  bo  a  minimum. 

Alls.  The  axis  must  be  parallel  to  the  major  axis,  and 
bisect  the  semi-minor  axis. 

11.  Find  the  centre  of  percussion  of  ■\  cube  w'lich  rotate? 
about  an  axis  parallel  to  the  four  parallel  edges  of  the  cube, 
and  equidistant  from  the  two  nearer,  as  well  as  from  the 
two  farther  edges.  Let  'ia  bo  a  side  of  the  cube,  and  let  ■:. 
be  the  distauce  of  the  rotation-axis  fn  m  its  centre  of 
gravity. 

I  =  c  ■\-  -5-,  where  /  is  the  distance  from  the  rota- 


/1ns. 


3c' 


tion-iixis  to  the  centre  of  percussion. 

12.  Find  .the  centre  of  ix^rcussion   of  a  sphere  which 
rotates  about  an  axis  tangent  to  its  surface. 

A,h9.  I  ss  \a. 

'  13.  Tift  the  body  in  Art.  243,  be  u  sphere  whoso  radius  is 
17  ins.  and  weight  1200  lbs. ;  let  it  be  moved  by  a  weight 
of  260  Iba  atf'"^hcd  to  a  cord,  wound  round  a  wheel  whose 


KXAMPLES. 


479 


quare,  each  sido 
,C3  about  one  of 
imple  equivalent 
.  f  Vi^T"  P. 

it  swings  about 
the  other  Ifitus 
BciUation;  prove. 

istapce  from  one 
ibout  which  the 
the  length  of  the 

centre  of  gravity 


ptic  lamina  must 
:)e  a  minimum. 
3  major  axis,  and 

lube  w'lich  rotates 
edges  of  the  cube, 
well  as  from  the 
10  cube,  and  let  :7 
«  m  its  centre  of 

ice  from  *;he  rotsi- 

F  a  sphere  which 
ce. 

A, IS.  I  =  \a. 

ere  whose  radius  is 
noved  by  a  weight 
nd  a  wheel  whose 


revolutions  of  the 
Ans.  5b  V. 


radius  Is  15  ins.;  And  the  number  of 

sphere  in  10  seconds,    (g  =  33.) 

•    14.  Let  the  body  in  Art.  243  be  a  sphere  of  radius  8  ins. 

and  weight  500  lbs. ;  let  it  be  moved  by  a  weight  of  10(^  lbs. 

attached  to  a  cord  wound  round  a  wheel  whose  radius  is 

6  in.;   find  the  number  of  revolutions  of  the  sphere  in 

5  seconds,     {ff  =  32^.)  4»s.  28.09. 

15.  In  Art  244,  let  the  weight  i^  =  40  lbs.,  W  =  100 
lbs.,  w  =  12  lbs.,  and  w'  =  0  lbs.;  and  let  E  and  r  be 
12  ins.  and  7  ins. ;  required  (1)  the  space  T>assed  over  by  P 
in  16  sees,  if  it  starts  from  rest,  and  (2)  the  tensions  T  and 
T'  of  the  cords  supporting  P  and  W.     {(/  =  32). 

Ans.  (1)  926.5;  (2)  T  =  49.04  lbs.,  T'  =  86.81  lbs. 

16.  In  Art.  344,  let  the  weight  P  -  25  lbs.,  W  -  60 
lbs.,  w  =  6  lbs.,  and  -lo'  =  2  lbs. ;  and  let  R  and  r  be 
8  in.  and  3  in.;  required  (1)  tlie  space  passed  over  by  P  in 
10  sees,  if  it  starts  from  rest,  and  (2)  the  tensions  T  and 
T'  of  the  cords  supporting  P  and  W.    {g  =  32|.) 

Ans.  (1)  109.92  ft;  (2)  T  =  23.29  lbs.;  T'=  G1.54  lbs. 

17.  In  Art  245,  let  the  body  be  a  sphere  whose  radius 
is  3  ft,  whose  weight  is  800  lbs.,  and  the  distance  of  whose 
centre  from  the  axis  is  9  ft. ;  and  let  F  he  a  force  of  60  lbs. 
acting  at  the  end  of  an  arm  12  ft  long;  find  (1)  the  num- 
ber of  revolutions  which  the  body  will  make  about  the 
axis  in  12  min.,  and  (2)  the  time  of  one  revolution. 
{g  =  32.)  Ans.  (1)  14043.6  ;  (2)  6.07  sees. 

18.  In  Ex.  17,  let  the  radius  =  one  foot,  the  weight  = 
100  lbs.,  the  distance  of  centre  from  axis  =  5  ft.,  and 
F  ■=  25  lbs.  acting  at  end  of  arm  8  ft  long;  find  (1)  the 
number  of  revolutions  which  the  body  will  make  about  the 
axis  in  5  min.,  and  (2)  the  time  of  one  revolution. 
(<,  =  32|.)  Ans.  (1)  18139.09  ;  (2)  2.23  sees. 

19.  If  the  body  in  Art  247  be  a  homogeneous  sphei-e, 
the  s*.riiig  being  round  the  circumference  of  a  great  circle, 


tmme&am^sss 


3S£££S 


480 


EXAMPLB8. 


find  (1)  the  angular  velocity  just  after  the  impulse,  and  (2) 
the  impulsive  tension.  ^       Sy 


2"  A  bar,  I  feet  long,  falls  vertically,  retaining  its  hori- 
zon^l  position  till  it  strikes  a  fixed  obstacle  at  one-quarter 
the  length  of  the  bar  from  the  centre  ;  find  (1)  the  angu- 
lar velocity  of  the  bar,  (2)  the  linear  velocity  of  its  centre 
just  after  the  impulse,  and  (3)  the  impulsive  force,  the 
velocity  at  the  iustan .  of  the  impulse  being  v. 

12« 

21.  A  bar,  40  ft.  long,  falls  through  a  vortical  height  of 
60  ft.,  retaining  its  horizontal  position  till  one  end  strikes 
a  fixed  obstacle  60  ft.  above  the  ground  ;  find  (1)  its  anjju- 
lar  velocity,  (2)  the  linear  velocity  of  its  centre  just  after 
the  impulse ;  (3)  the  number  of  revolutions  it  will  make 
before  reaching  the  g;ound,  (4)  the  whole  time  of  falling 
to  the  ground,  and  (5)  i\&  linear  velocity  on  reaching  the 
ground. 
^  Ann.  (1)  2.12;   (2)  42.43;   (3)  0.345;    (4)  2.79;    (5) 


I  ' 


1 


npulso,  and  (2) 

aining  its  hori- 
I  at  one-quartor 
(1)  the  angu- 
ty  of  its  centre 
Isive  force,  the 


|v ;  (3)  Htnv. 

rtical  height  of 
one  end  strikes 
id  (1)  its  angu- 
entre  just  after 
18  it  vill  make 
I  time  of  falling 
)U  reaching  the 

(4)  2.79;   (5) 


CHAPTER    VHI. 

MOTION    OF    A    SYSTEM    OF    RIGID   BODIES  IN   SPACE. 

248.  The  EqnationB  of  Motion  of  a  System  of 
Rigid  Bodies  obtained  by  D'Alemberf s  Principle.— 

Let  {x,  y,  z)  be  the  position  of  the  particle  m  at  the  time  / 
referred  to  any  set  cf  rectangular  axes  fixed  in  space,  and 
X,  r,  Z,  the  axial  components  of  the  impressed  accelera- 
ting forces  acting  on  the  same  particle.    Then  -ttj  ,  j^f  >  ;^. 

are  the  axial  components  of  the  accelerations  of  the  parti- 
cle ;  and  by  D'Alembert's  Principle  (Art.  235)  the  forces, 

«(x-'^).  ,„(K-g).  ,«(z-f;). 

acting  on  m  together  with  pimilur  forces  acting  on  every 
f  article  of  the  system,  are  in  equilibrium.  Hence  by  the 
principles  of  Statics  (Art  G5)  we  have  the  following  six 
equations  of  motion  : 


(1) 


(2) 


21 


482 


TRAXSLATION  AND  ROTATIOIT. 


w\ 


\M 


m 


By  means  of  these  six  equations  the  motion  of  a  rigid 
body  acted  on  by  any  finite  forces,  may  be  determined. 
They  lead  immediately  to  two  important  propositions,  one 
of  which  enables  us  to  calculate  the  motion  of  translation 
of  the  body  in  space ;  and  the  other  the  motion  of  rotation. 

L49.  Independe&oe  of  the  Motion  cf  Translation 
of  ths  Centre  of  Gravity,  and  of  Rotation  about  an 
Axis  Passing  through  it.- -Let  (i,  y,  i)  be  the  position 
of  the  centre  of  granty  of  the  body  at  the  time  t,  referred 
to  fixed  axes,  {x,  y,  z)  the  position  of  the  particle  m  referred 
to  the  same  axes,  («',  y,  z')  the  position  of  m  referred  t^  a 
Bystcui  of  axes  passing  through  the  centre  of  gravity  and 
parallel  to  the  fixed  axes,  and  M  the  whole  mass.    Then 

1.         x=zi  +  x',    y  =  y  +  y',    z  —  l  +  z'.  (1) 

Since  the  origin  of  the  movable  system  is  at  the  centre  of 
gravity,  we  have  (Art.  59) 

Ima;'  =  7.my'  =  Smz'  =  0 ;  (2) 

^    <ftB'       ^,     «Py'       ^     (Pz' 


Ki; 


Also     "Lmas  =  Mx,     ^my  =  My,  Swa  =  MZ; 


^"*5^  =  ^5^'  ^""dt^ 


M 


df»' 


d>z      „<?>« 


Substituting  these  values  in  (1)  of  Art.  248,  wo  have 


dr 


(4) 


ON. 

notion  of  a  rigid 
be  determined, 
propositions,  one 
n  of  translation 
otion  of  rotation. 

cf  Translation 
;aiion  about  an 

)  be  the  position 
e  time  /,  referred 
(article  m,  referred 
I  m  referred  t.:.  » 
re  of  gravity  and 
3  maas.    Then 

5  +  «'.  (1) 

is  at  the  centre  of 

0;  (2) 

=:a  (3) 

.  248,  we  have 


(4) 


MOTION   OF  A    BO  or. 


488 


These  three  eqnations  do  not  contain  the  co-ordinatos  of 
the  point  of  apphcation  of  the  forces,  and  are  the  same  as 
those  which  would  be  obtained  for  the  motion  of  the 
contre  of  gravity  supposing  the  forces  all  applied  at  that 
point.    Hence 

The  motion  of  the  centre  of  gravity  of  a  system  acted  on 
by  any  forces  is  the  same  as  if  all  the  mass  were  collcded  at 
the  centre  of  gravity  and  all  the  forces  were  applied  at  that 
point  parallel  to  t/ieir  former  directions. 

2.  Differentiating  {] )  twice  we  have 

W  ~  dt^  ■*"  rf^«'    di^  ~  dt»  "^  dt^' 

dt*  ~  dt^^  dP' 

Substituting  these  valras  in  the  first  of  equations  (2)  of 
Art.  248,  we  have 

2m[(y+>')Z-(5  +  On 
Performing  the  operations  indicated  we  get 


Uii>- 


f 


II 

11 


484 


TRAySLAriON  AND  ROTATION. 


I 


Omitting  the  Ist,  2d,  4th,  5th,  Gth,  and  8th  terms  which 
vanish  by  reason  of  (2),  (3),  and  (4),  .we  have 

similarly  from  the  other  two  equations  o'  (2)  \  e  have/ 


(•^) 


j  =  I.mix'r-y'X). 

These  three  equations  do  not  contain  the  co-ordinates  of 
the  centre  of  gravity,  and  are  exactly  the  equations  we 
would  have  obtained  if  we  had  regarded  the  centre  of 
gravity  as  a  fixed  point,  and  taken  it  as  the  origin  of 
moments.     Hence 

ne  motion  of  a  body,  acted  on  by  any  forces,  about  its 
centre  of  gravity  is  the  samp,  as  if  the  centre  of  gravity  were 
fixed  and  the  same  forces  acted  on  the  body.  That  is,  from 
(4)  the  motion  of  translation  of  the  centre  of  gravity  of  the 
body  is  independent  of  its  rotation  ,•  and  from  (5)  the  rota- 
tion of  the  body  is  independent  of  the  translation  of  its 
centre. 

These  two  important  propositions  are  called  respectively, 
the  principles  of  the  conservation  of  the  motions  of  transla- 
tion and  rotation. 

Sen.— -By  the  first  principle  the  problem  of  finding  the 
motion  of  the  centre  of  gravity  of  a  system,  however  com- 
plex the  system  may  be,  is  reduced  to  the  problem  of 
finding  the  motion  of  a  single  particle.  By  the  second 
principle  the  problem  of  finding  the  angular  motion  of  a 
fiw  body  in  space  is  reduced  to  that  of  determining  the 
motion  of  that  body  about  a  fixed  jmint 


ON. 

3th  terms  which 
ive 


•2)  y  e  have^ 
x'Z), 


CONSBltVATION  OF  CENTRE  OF  ORAVITT. 


485 


(&) 


y'X). 


Rem.— In  using  the  first  principle  it  should  be  noticed 
that  the  impressed  forces  are  to  be  applied  at  the  centre  of 
gravity  parallel  to  their  former  directions.  Thus,  if  a  rigid 
body  be  moving  under  the  influence  of  a  central  force,  the 
motion  of  the  centre  of  gravity  is  not  generally  the  same 
us  if  the  whole  mass  were  collected  at  the  centre  of  gravity 
and  it  were  then  acted  on  by  the  same  central  force.  What 
the  principle  asserts  is,  that,  if  the  attraction  of  the  central 
force  on  each  element  of  the  body  be  found,  the  motion  of 
the  centre  of  gravity  is  the  same  as  if  these  forces  were 
applied  at  the  centre  of  gravity  parallel  to  their  original 
directions. 


le  co-ordinates  of 
le  equations  we 
d  the  centre  of 
as  the  origin  of 

forces,  about  its 
e  of  gravity  were 
t.  That  is,  from 
I  of  gravity  of  the 
from  (5)  the  rota- 
ranslation  of  its 

lied  respectively, 
Hions  of  transltt- 

n  of  finding  the 

m,  however  com- 

the  problem  of 

By  the  second 

ular  motion  of  a 

determining  the 


250.  The  Principle  of  the  Conservation  of  the 
Centre  of  Gravity.— Suppose  that  a  material  system  is 
acted  on  by  no  other  forces  than  the  mutual  attractions  of 
its  parts ;  then  the  impressed  accelerating  forces  are  zero, 
which  give 

2X=SF=SZ=0; 

thereiore  from  (4)  of  Art.  249,  we  get 


0.    ^  =  0; 


^^  =  r„cos«, 


^  =  »o  COS  ft    ^  =  «o  COS  y. 


(1) 


where  i;„  is  the  velocity  of  the  centre  of  gravity  when 
f  =  0,  and  «,  ft  y,  are  the  angles  which  its  direction  makes 
with  the  axes.  Therefore,  calling  v  the  valocity  of  the 
centre  of  gravity  at  the  time  /,  we  liave 


V=:^J'■ 


iV?  +  dy^  +  d? 


dC^ 


=  v, 


0* 


(3) 


lohich  is  evidently  coiistunt. 


486 


CONSERVATION  'OF  AREAS. 


If  Vj  =  0,  the  centre  of  gravity  remainx  at  rest. 
Integrating  (1)  we  get 

i  =z  v^t  COB  a  ■+■  a,    y  =  v^t  008/3  +  4, 

i  =  v„f  COS  y  +  c ; 

a  —  a  _ y  —  b _  z  —  c 
cos  a  "~  cos  /3  ~  cos  y 


(3) 


(ff,  b,  c)  being  the  platje  of  the  centre  of  gravity  of  the 
system  when  /  =  0.  As  (3)  ar:  the  equations  of  a  straight 
line  it  follows  that  the  motion  of  the  centre  of  gravity  is 
rectilinear. 

Hence  token  a  material  system  is  in  motion  under  the 
action  of  forces,  none  of  which  are  external  to  the  system, 
then  the  centre  of  gravity  moves  uniformly  «i  a  straight  line 
or  remains  at  rest. 

REM.^Thns  the  motion  of  the  centre  of  gravity  of  a 
system  of  particles  is  not  altered  by  their  mutual  collision, 
Avhatever  degree  of  elasticity  they  may  have,  because  a 
reaction  always  exists  equal  and  opposite  to  the  action.  If 
an  explosion  occurs  in  a  moving  body,  whereby  it  is  broken 
into  pieces,  the  line  of  motion  and  the  velocity  of  tlie 
centre  of  gravity  of  the  body  are  not  changed  by  the 
explosion  ;  thus  the  motion  of  the  centre  of  gravity  of  the 
earth  is  unaltered  by  earthquakes  ;  volcanic  explosions  on 
the  moon  will  not  change  its  motion  in  space.  The  motion 
of  the  centre  of  gravity  of  the  solar  system  is  not  affected 
by  the  mutual  and  reciprocal  action  of  its  several  members ; 
it  is  changed  only  by  the  action  of  forces  external  to  the 
system. 

(     251.  The  Principle  of  the  Conservation  of  Areas.— 

If  X,  y  be  the  rectangular,  and  r,  0  the  polar  co-ordiuatcs 
of  u  particle,  we  have 


1 


I. 

at  rest. 

113  +  *, 


m 


[)f  gravity  of  the 
tions  of  a  straight 
ntre  of  gravity  is 

motion  under  the 

'tial  to  the  system, 

in  a  straight  line 

e  of  gravity  of  a 
mutual  collision, 
have,  because  a 
to  the  action.  If 
loreby  it  is  broken 
le  velocity  of  the 
;  changed  by  the 
e  of  gravity  of  the 
;anic  explosions  on 
pace.  The  motion 
em  is  not  affected 
6  several  members ; 
cea  external  to  the 


ation  of  Areas. — 

3  polar  co-ordinates 


coNsesvATioy  of  axsas. 


r» co8»  e  ^  (tan  e)  =  /^^. 
dt^        '  dt 


487 


(1) 


Now  \i/^dO  is  the  elementary  ai-ea  described  round  the 
origin  in  the  time  dt  by  the  projection  of  the  radius  vector 
of  the  pai-ticle  on  the  plane  of  xy,  (Art.  182.)  If  twice 
this  polar  area  be  multiplied  by  the  mass  of  the  particle, 
it  is  called  the  area  conserved  by  the  particle  in  the  time  dt 
round  the  axis  of  z.    Hence 

is  called  the  area  conserved  by  the  system. 

Let  dAx,  dAy,  dAg  be  twice  the  areas  described  by  the 
projections  of  the  radius  vector  of  the  particle  m  on  the 
planes  olyz,  zx,  xy,  respectively ;  then  from  (1)  we  have 

^     I  dy         dx\       ^    dAg 
and  differentiating  we  get 


„    /  d^/         €Px\       „    d^A, 


(2) 


If  the  impressed  accelerating  forces  are  zero  the  first 
member  of  (2)  is  zero,  from  (5)  of  Art.  24i/;  therefore  the 
second  member  is  zero.    Hence 


v^.^-^'  A. 


dt^ 


aM 


488 


CONSEBVATTON  OF   VIS    VIVA. 


Similarly    .    1?»  -~  —  0,    Sw  -^  =  0 ; 
and  therefore  by  integration 

at  dt  at 


«", 


h,  h',  h"  being  constants. 

.  • .    ^mAx  =  ht,    ^mAy  =  h't,    ImAg  —  h"t ; 

the  limits  nf  integration  being  snch  that  the  areas  and  the 
time  begin  simultaneously.  Thus,  the  sum  of  the  products 
of  the  mass  of  every  particle,  and  the  Drojection  of  the  area 
described  by  its  radius  vector  on  each  co-ordinate  plane, 
varies  as  the  time.  This  theorem  is  called  the  principle  of 
the  conservation  of  areas.    That  is, 

When  a  material  system  is  in  motion  under  the  action 
offerees,  none  of  which  are  external  to  the  systetn,  tJten  the 
sum  of  the  products  of  the  mass  of  each  partich  by  the  pro- 
jection,  on  any  plane,  of  the  area  described  by  the  radius 
vector  of  this  particle  measured  from  any  fixed  point,  varies 
as  the  time  of  motion. 

252.  Conservation  of  Vis  Viva  or  Energy.'*'— Let 

{x,  y,  z)  be  the  place  of  the  particle  m  at  the  time  t,  and 
let  X,  Y,  Z  be  the  axial  components  of  the  impressed 
accelerating  forces  acting  on  the  particle,  as  in  Art.  348. 
Tlie  axial  components  of  the  efEective  forces  acting  on  the 
same  particle  at  any  time  /  are 

dhi        d^        ^ 

"^d?'   "'Sp'   "^di^- 

'  If  the  efEective  forces  on  all  the  particles  be  reversed, 

*  See  Art.  189. 


"hU 


1 


h"t', 


e  areas  and  the 
of  the  products 
stion  of  the  area 
i-ordinate  plane, 
the  principle  of 

under  the  action 
system,  tJien  the 
iiclo  by  the  pro- 
id  by  the  radius 
Ixed  point,  varies 

Energy.*— Let 
b  the  time  t,  and 
f  the  impressed 
I,  as  in  Art.  248. 
3s  acting  on  the 


cles  be  reversed, 


?«r?Ri 


IWWIiyi«^l|JMilMPM;.l  iW^i'.tpM^JttlHMIIfm   J- 


CONSSRVATION  OF  VIS    VIVA. 


489 


they  will  be  in  equilibrinm  with  the  whole  gronp  of  im- 
pressed forces  (Art.  236).  Hence,  by  the  principle  of 
virtual  velocities  (Art.  104),  we  have 

.».[(x-*)^^(r-g).,  +  (z-g).«]=„.,:, 

where  dx,  6y,  dt  are  any  smaU  arbitrary  displacements  of 
the  particle  m  parallel  to  the  axes,  consistent  with  the  con- 
nection of  the  parts  of  the  system  with  one  another  at  the 
time  t. 

Now  the  spaces  actually  described  by  the  particle  m  dur- 
ing the  instant  after  the  time  t  ijarallel  to  the  axes  are 
consistent  with  the  connection  of  the  parts  of  the  system 
witli  each  other,  and  hence  we  may  take  the  arbitrary  dis- 
placements, Sx,  6y,  6z,  to  be  respectively  equal  to  the 

actual  displacements,  ^ 6t,  ^  6t,  j^  6t,  of  the  particle.* 
Making  this  substitution,  (1)  becomes 

^"^  w  dt  ^  d(^  dt'^  m^  It) 


-(^f+4f+4)- 


Integrating,  we  get 

£»««;»  -  27«v„»  =  2Sm  /  {Xdx  +  Ydy  +  Zdz),     (2) 

vhere  v  and  v^  are  the  velocities  of  the  particle  m  at  the 
times  t  and  t^. 

The  first  member  of  (2,  is  twice  the  vis  viva  or  kinetic 
euL  '^  of  the  system  acquired  in  its  motion  from  the  time 
^0  to  the  time  t,  under  the  action  of  the  given  forces. 


,  *  '^i^'  /"•  t'"»>"'8'» ««  ^  •«»»  eqoal  to  <te,  yel  the  nwto  of  &e  to  dit  is  equal  lo  tho 
rutio  01  it  Unit. 


ii 


iif 


490 


OONBEHVATION  OF  VIS  VIVA. 


Tho  second  member  expresses  twice  the  work  done  by 
these  forces  in  tho  same  time  (Art  189). 

If  the  second  member  of  (2)  be  au  exact  differential  of  a 
function  of  x,  y,  z,  so  that  it  equals  df{x,  y,  z) ;  then  tak- 
ing the  definite  integral  between  tho  limits  z,  y,  z  and  x^, 
y^i  Xq)  corresponding  to  t  and  t^,  (2)  becomes 

2mv»  -  £mvo»  =  2/(«,  y,  z)  -  Zfix^,  y^,  z^).     (3) 

Now  the  second  member  of  (2)  is  an  exact  differential  so 
far  as  any  particle  m  is  acted  on  by  a  central  force  whose 
centre  is  fixed  at  (a,  b,  c),  and  which  is  a  function  of  the 
distance  r  between  the  centre  and  {(jr.,  y,  r)  the  place  of  m. 
Thus,  let  P be  the  central  force  =/{r),  say;  then 


X  —  a 


fir),    r  = 


:r_y-i 


fir), 


Z=^'-^fir)i 


r»  =  («  -  a)«  +  (y  -  t)»  +  (z  -  c)»; 
, • ,    rdr  =  {x  —  a)dx  +  (y  —  b)dy  +  {z  —  c)dzi 
.'.    m  (Xdx  +  Ydy  +  Zdt)  =  m/{r)  dr-, 


which  is  an  exact  differential ; 
second  member  of  (2),  it 


substituting  this  in  the 


2m 


rfir)dr, 


where  the  limits  r  and  r,  correspond  to  t  and  /j. 

Also,  the  second  member  of  (2)  is  an  exact  differential, 
so  far  as  any  two  particles  of  the  system  are  attracted 
towards  or  repelled  from  er oh  other  by  a  force  which  varies 
as  the  mass  of  each,  and  is  a  function  of  the  distance 
between  them.  Let  m  and  m'  be  any  two  particles ;  let 
(a;,  y,  z),  («',  y',  z')  be  their  places  at  the  time  t ;  r  their 
distance  apart;  P  =  f{r),  the  mutual  action  of  the  unit 
mass  of  each  particle.    Then  the  whole  attractive  force  of 


'r'i£«ne#a:<,««3i»i$^&wM«%i#^.^«««£. 


ijiiiriill,,. 


work  done  by 

differential  of  a 
y,  z) ;  then  tak- 
i  x,y,z  and  x^, 
nes 

«o»  yo'  «o)-      (3) 

MJt  differential  so 
ntral  force  whose 
\  function  of  the 
r)  the  place  of  m. 
say;  then 


z  =  ^VW; 


-  (z  —  c)dz', 

f(r)dr', 

ating  this  ui  the 


t  and  /g* 

exact  differential, 
rstem  are  attracted 
I  force  which  varies 
ion  of  the  distance 

two  particles ;  let 
he  time  t ;  r  their 

action  of  the  unit 
e  attractive  force  of 


"Vg".y'™V!ty'gT!*;gwT>~y«!»^ggi'i;«^^^^ 


CONSERVATION  OP  VIS   VIVA, 


491 


m  on  m'  is  Pm,  and  the  whole  attractive  force  of  »? '  on  to 
is  Pm' ;  and  we  have 


X 


—   *M  ^ iL 


Z-lif 


X=.m~LF,     Y=m^-=^P,    Z=m^-P; 


r=-m^^P,    Z'=z. 


m J 

r 


r       ■  r 

Also     r*  =  («  -  «')»  +  (y  -  y')*  +  («  -  «')'• 
Therefore  for  these  two  particles,  we  have    • 
m  (Xdx  +  Ydy  +  Zdz)  +  to'  (X'da;'  +  F'rfy'  +  Z'dz') 

=  ~^  [(«  -  *')  ('^a;  -  rfo;')  +  (y  _  y')  (rfy  _  rfy') 

+  {z-  z')  {dm  —  </«')] 
=  tnm'f{r)dr; 

which  is  an  exact  differential.  The  same  reasoning  applied 
to  every  two  particles  in  the  system  must  lead  to  a  similar 
result ;  so  that  finally  the  second  member  of  (2) 


=  Stoto'  //(r) dr, 


where  the  limits  r  and  r,  correspond  to  t  and  <„,  so  that 
the  integral  will  be  a  function  solely  of  the  initial  and  final 
co-ordinates  of  the  particles  of  the  system. 

Hence,  wJisn  a  material  system  is  in  motion  under  the 
action  of  forces,  none  of  which  are  external  to  the  system, 
then  the  chaise  of  the  vis  viva  of  the  system,  in  passing 
from  one  position  to  another,  depends  only  on  the  two  posi- 
tions of  the  system,  and  is  independent  of  the  path  described 
by  each  particle  of  the  system. 

This  theorem  is  called  the  principle  of  the  conservation  of 
vis  viva  or  energy. 


PntNClPLE  OF  VIS  VIVA. 

Gob.  1. — If  a  system  be  under  the  action  of  no  external 
forces,  we  ha?e  X  =  V  =  Z  =  0,  and  hence  the  vis  viva 
of  the  system  is  constant. 

OoB.  9.. — lict  gravity  bo  the  only  force  acting  on  the 
system.  Let  the  axis  of  c  be  vertical  and  positive  down- 
wards, then  we  have  X  =  0,  Y  =  0,  Z  =  g.  Hence  (3) 
becomes 


But  if  e  and  s,  are  the  distences  !W)m  the  plane  of  xy  to  the 
centre  of  gravity  of  the  system  at  the  times  t  and  t^,  and  if 
M  is  the  mass  of  the  system,  we  have 


Mi  —  I.mz,    ilfio  =  Xm«o; 


(4) 


That  is,  the  increase  of  vis  viva  of  the  system  depends  only 
on  the  vertical  distance  over  which  the  centre  of  gravity 
passes  :  and  therefore  the  vis  viva  is  the  same  whenever  the 
centre  of  gravity  passes  through  a  given  horizontal  plane. 

Rem. — The  principle  of  vis  viva  was  first  used  bj  Hayghens  in 
M»  doternilnation  of  tuo  centre  of  oscillation  of  a  body  (Art.  237, 
llcm.). 

Tlio  ndvantag«  of  tliie  principle  ia  that  It  gives  at  onro  a  rnlution 
between  the  velocities  of  the  bodias  considered  and  the  coordinates 
which  determine  their  (tositions  In  space,  so  that  when,  from  the 
nature  of  the  problem,  the  position  of  all  the  bodies  may  bo  made  to 
depend  on  one  variable,  the  equation  of  vis  viva  is  sufficient  to  deter- 
mine the  motion. 

Suppose  a  weight  m^  to  be  ])lBced  nJ.  any  height  A  above  tho  nur- 
face  of  tlie  earth.  As  it  falls  through  a  height  c,  the  force  of  gravity 
does  work  which  ia  measured  by  mgt.  Tho  weight  has  acquired  % 
Velocity  V,  and  therefore  its  vis  viva  is  imc'  wldch  is  wjual  to  t/ig$ 
(Art.  21  ).  If  the  weiglit  falls  through  the  remainder  of  tho  height 
/(,  gniviiy  diKw  nuirn  wtirli  wliich  iu  monsuKHl  by  mff  (A  —  «).  Wliun 
tlie  weight  huu  nuclii'd  tli><  ground,  it  huu  fallen  aa  far  as  the  ciruuui- 


)f  no  external 
36  the  vis  viva 


acting  on  the 
positive  dowii- 
g.    Hence  (2) 


)• 


me  of  xy  to  the 
!  and  t^,  and  if 


,). 


(4) 


m  depends  only 
ntre  of  gravity 
m  whenever  the 
izontal  plane. 

d  bj  Huyghens  in 
a  body  (Art.  337, 

at  onco  a  relation 
id  tbe  coordinates 
»t  when,  from  the 
38  may  be  made  to 
I  sufficient  to  deter- 

it  h  above  tl:n  mr- 
the  force  of  gravity 
gbt  has  acquirwl  a 
ich  is  wjual  to  »/»</» 
inder  of  the  height 
mt/(h-s).  When 
a  far  as  the  circuui- 


"li 


COMPOSITION  OF  ROTATIONS. 


403 


stance.!  of  the  case  permit,  and  gravity  has  done  work  which  is  meas- 
ured by  ii.^h,  and  can  do  no  more  work  until  the  weight  has  been 
lifted  up  again.  Hence,  thionghout  the  motion  when  the  weight  has 
descended  through  any  jpaoe  (s,  its  vis  viva,  \mx?{=mgt),  together 
with  the  work  that  can  be  done  during  the  rest  of  the  descent, 
mg  (h  —  e),  is  constant  and  equal  to  mgk,  the  work  done  by  gravity 
daring  the  wliole  descent  h. 

It  we  complicate  the  motion  by  making  the  weight  work  some 
machine  during  its  dewcent,  the  wamo  tlieorem  ia  still  true.  The  vis 
viva  of  the  weight,  when  it  has  deflcen(*ed  any  space  i:.  Is  equal  to  the 
work  mgz  which  has  bren  done  by  gravity  during  this  descent,  dimin- 
ished by  the  work  done  on  the  machine.  Hence,  as  before,  the  vis 
viva  together  with  the  diflf&rence  between  the  work  done  by  gravity 
•  and  that  done  on  the  machine  during  the  remainder  of  the  descent  is 
constant  and  equal  to  the  excess,  of  the  work  done  by  gravity  over 
that  done  on  the  machine  during  tho  whole  descent.  (See  Routli's 
Rigid  Dynamics,  p.  270.) 

253.  Composition  of  Rotations.— It  is  often  neces- 
saiy  to  compound  rotations  about  axes  which  meet  at  u 
point.  When  a  body  is  said  to  have  angular  velocities 
about  three  diiferent  axes  at  the  same  time,  it  is  only  meant 
that  the  motion  may  be  determined  as  follows :  Divide  tho 
whole  time  into  a  number  of  infinitesimal  intervals  each 
equal  to  dt.  During  each  of  those,  turn  the  body  round 
the  three  axes  successively,  through  angles  ^idi,  Wgdt,  u^dt. 
The  result  will  be  the  same  in  whatever  order  the  rotationn 
take  place.  Tho  flnal  displacement  of  the  body  is  tho 
diagonal  of  the  parallelepiped  described  on  these  throe  linos 
as  sides,  and  is  therefore  independent  of  the  order  of  the 
rotations.  Since  then  the  three  successive  rotations  are 
quite  independent,  thoy  may  be  s»»'..I  to  take  place  simul- 
taneously. 

Hence  we  infer  that  angular  velocities  and  angular  acnel- 
erationi*  may  be  compounded  and  resolved  by  the  saiuo 
rules  and  in  the  same  way  as  if  they  were  linear.  Thus,  an 
angular  velocity  w  about  any  given  axis  may  be  resolved 
into  two,  w  cos  a  and  w  sin  «,  about  axes  at  right  angles  to 


494 


MOTION  OF  A  RIGID  BODY. 


oach  other  and  making  angles  a  and  ^  —  «  with  the  given 


axis. 


Also,  if  a  body  have  angnlar  yelocities  6)^,  &>,,  6),  about 
three  axea  at  right  angles,  they  are  to /ether  equivalent  to 
a  single  angalar  velocity  a»,  where  w  =  Vwi'-j-Wi'+Wj^ 
about  an  axis  inclined  to  the  given  axes  at  ungles  whose 


cosined  are  respectively  --,-*, 


-1. 


254.  Motion  of  a  Rigid  Body  referred  to  Fixed 

Ax-  '■- — Let  U8  suppose  that  one  point  in  the  body  is  flxed. 
I.,:  .....  point  be  taken  as  the  origin  of  co-ordinates,  and 
let  tlie  axes  OX,  OY,  OZ  be  any  directions  fixed  in  space 
and  at  right  angles  to  one  another.  The  body  at  the  time 
t  is  turning  about  some  axis  of  instantaneous  rotation 
(Art.  240).  Let  its  angular  velocity  about  this  axis  be  <•>, 
and  let  this  be  resolved  into  the  angular  velocities  <>>,,  6>„ 
fa>,  about  the  co-oixlinat^  axes.     It  is  required  to  find  the 

roaolved  linear  velocities,  jft  ^■>-jf>  parallel  t)  the  axes  of 

co-ordinates,  of  a  particle  wi  at  the  point  P,  {x,  y,  z),  in 
terms  of  the  angular  velocities  about  the  axes. 

These  angular  velocities  are  sup- 
posed positive  when  they  tend  the 
same  way  round  the  axes  that 
positive  CO'  .;  -  *'jnd  in  Statics 
(Art.    06).         'i'  i   the    positive 


direction^  Oi  ■•  -  ,,  w,  are  re- 
spectively froi.  ;^  to  z  about  x, 
from  «  to  a;  about  y,  and  from  x 
to  y  about  t ;  and  those  negative 
which  act  in  the  opposite  direc- 
tions. 

Tjot  us  determine  the  velocity 
of  P  parallel  to  the  axis  of  t.     Let  PN  bo  the  ordinate  z, 


na.ioo 


with  the  given 

,  w,,  w,  about 
•  equivalent  to 

A  angles  whose 


red  to  Fixed 

e  body  is  fixed. 
>-ordinateB,  and 
i  fixed  in  space 
ody  at  tht  time 
ineons  rotation 
this  axis  be  <•>, 
jloeities  <•>,,  w», 
red  to  find  the 

el  i  >  the  axes  of 

P,  («,  y, «)» »» 

Les. 


ria.iQO 
0  the  ordinate  z, 


AXIS  OF  INaTAJfTANSOUS  BOTATION. 


495 


and  draw  PM  perpendicular  to  the  axis  of  x.  The  velocity 
of  P  doe  to  rotation  about  OX  is  u^PM,  Resolving  this 
parallel  to  the  axes  of  y  and  z,  and  reckoning  those  linear 
velocities  positive  which  tend  from  the  origin,  and  vice 
verm,  we  have  the  velocity 

along        MN  =  —  t^^PM cos  NPM  =—«,«; 
and  along      NP  =  UiPM  sin  JVPJf  =  w,y. 

Similarly  the  velocity  dne  to  the  rotation  about  OF  par- 
allel to  OX  is  iifZ,  and  parallel  to  OZ  is  —  u^x.  And  that 
due  t3  the  rotation  about  OZ  parallel  to  CX  is  —  «jy,  and 
parallel  to  OF  is  <o^x. 

Adding  together  those  '^looities  which  are  parallel  to 
the  same  axes,  we  have  for  the  velocities  of  P  p^trallel'to 
the  axes  of  x,  y,  and  z,  respectively. 


dx 
It 


■  =  6).2 


dy 


"sy. 


"i«» 


(1) 


dz 
^^u>,y-u>,x. 


255.  Azi«  of  Xnatantaneoiu  Rotatioa-^Every  par- 
ticle in  the  axis  of  instantaneous  rotation  is  at  rest  relative 
to  the  origin ;  hence,  for  these  particles  each  of  the  first 
members  of  (1)  in  Art.  264,  will  reduce  to  zero,  and  wc 
have 


«i«  —  w«y  =  0, ' 
w,«  "  «,«  =  0, 


(1) 


496 


ANOULAK    VELOCITY. 


which  are  the  equations  of  the  axis  of  instantaneous  rota- 
tion, the  third  equntion  being  a  tiecessary  consequence  of 
the  first  tvio;  hence. 


(li.  u. 

«  =  --«,    y  =  -a  « ; 


u. 


w. 


(2) 


that  is,  tho  instantaneous  axis  is  a  straight  line  passing 
through  the  origin  which  is  at  rest  at  the  instant  con- 
sidered ;  and  the  whole  body  must,  for  the  instant,  rotate 
about  this  line. 

Cob. — Denote  by  a,  /3,  y  the  angles  which  this  axis 
makes  with  tho  co-ordinate  axes  »,  y,  «,  respectively, 
then  (Anal-  Geom.,  Art.  175)  we  have 


cos  «  = 


«4 


Vw,»  +  Wg*  +  w,»' 


COS  |3  s= 


(t). 


V<V  +  Wg'-I-  w,« 


COS  y  = 


(0. 


V'wi»  + V  +  '^s*' 


which  gives  tht  position  of  the  instantaneous  axis  in  terms 
of  the  atigular  velocities  about  the  co-ordinate  axes. 

256.  The  Angular  Velocity  of  the  Body  abont  the 
Axis  of  Inatantaneous  Rotation. — The  angular  veloc- 
ity of  tho  body  about  this  axis  will  be  the  same  as  that  of 
any  single  particle  chosen  at  pleasui-e.  Lei  the  particle  be 
taken  on  the  axis  of  a; ;  if  from  it  wo  draw  a  perpendicular, 
p,  to  the  instantaneous  axis,  then  tho  distance  of  the  par- 
ticle from'  tho  origin  being  x,  we  have 


tntaneoua  rota- 
QQimquence  of 


(8) 


ht  line  passing 
le  instant  ^on- 
I  instant,  rotate 


vhich  this  axis 
«,  respectively, 


m  axis  in  terma 

te  0X68. 

Body  aboat  the 

le  angular  veloc- 
e  same  as  that  of 
}t  the  particle  be 
r  a  perpendicular, 
tance  of  the  par- 


SULER'a  XQUdTlOIfS, 


m 


p  =  XBm  a 


=  X  Vl  —  cos»  a  =  a;  A  /  —  '^i    "+•  "^ — 


Since,  for  this  particle,  y  =  0,  «  =  0,  we  have  from  (1) 
of  Art.  254,  for  the  absolute  velocity, 

and  hence,  for  the  angular  velocity  ?>,  we  have 


V  =  —  =  V«i*  +  w,*  +  w,», 

wAtcA  »«  /Ae  angular  telodty  required. 

257.  Euler'B  Equations. — To  determine  the  general 
equations  of  molio^n  of  a  body  about  a  fixed  point. 

Let  the  fixed  point  0  be  taken  as  origin  ;  let  {x,  y,  f)  be 
^he  place  of  any  jiarticle  m,  at  the  time  /,  referred  to  any 
rectangular  axes  fixed  in  spaoe,  a..  ^  let  Ox^,  Oyi,  Ozi  be 
the  rrincipal  axes  of  the  body  (Art.  231).  Piffereutiating 
(1)  of  Art.  254  with  respect  to  t,  we  have 

d?~'^~d~^~dt'^^*  <"»y  ~  ^^'^^  "■  "'  (<^8«~'^i«)j 

<Ptf  rfw.         dot.    ,        ,  .  .  V 

d^  ~'^'di  ~'^~d  ^^*  ^^**  ~  "»^^  ~  *^*  ^"'^  ~  "»^^' 

<P«         dill.         dii),  .        ,  ,  ,  V 

dfl-^~di         W'^'^^  ^"»*  ~  "»*^  ~  *"»  ^"•*  ~  ''•^^• 

Denoting  by  L,  M,  N,  the  first  terms  respectively  of  (2), 
(Art.  248),  and  substituting  the  above  values  of  ^  and  t^ 
in  the  last  of  these  equations,  wo  get 


4t)8 


BULSR'S  SqUATIONS. 


=  jv:    (1) 


The  other  two  eqnatioDB  may  he  treated  in  the  same  way. 

The  coefficients  in  thiq  equation  are  the  moments  and 
products  of  inertia  of  the  body  with  regard  to  axes  fixed  in 
space  (Art  224),  and  are  therefore  variable  as  the  body 
moves  about.  Let  Uj,,  tjy,  u,  be  the  angular  velocities  about 
the  principal  axes.  Since  the  axes  fixed  in  space  are  per- 
fectly arbitrary,  let  them  be  so  chosen  that  the  principal 
axes  are  coinciding  with  them  at  the  moment  under  con- 
sideration.   Then  at  this  moment  we  have  (Art  232), 

litnxy  =  0,    ^myz  =  0,    "Lmzx  =  0 ; 

also  6),  =  Wj,,  6)g  =  6)y,  0),  =  u, ;  and  likewise  -^  =  ~^^, 

etc.*    Hence,  denoting  by  ^,  B,  G,  the  moments  of  inertia 
about  the  principal  axes  (Art  231),  (1)  becomes 


''^-(^ 


B)  6)a,w^  =  N, 


in  which  all  the  coefficients  are  constants ;  and  similarly 
for  the  other  two  equations. 

Hence,  uniting  them  in  order,  and  retaining  the  letters 
b)j,  (i>„  fa),,  since  they  are  equal  to  Uo;,  (^,  (•>,,  the  three 


,  for  the  chanRM  In  the  two  angolar  Tclocltloe,  u,  and  <■<«,  dnrinir  a 


W  ~'  of 

tfiven  BDitll  time  after  the  axle  of  a),  colneide*  with  the  axis  of  x,  will  dUhr  only  by 
a  unantlly  which  depends  npon  the  angle  passed  throngh  by  the  axis  of  x,  dnring 
timt  given  khisII  time ;  the  dUferenoo  between  u,  and  oi*  will  therefore  be  an 
Inflnitesimal  of  the  second  order  and  thc.cfore  their  derlratWes  will  bo  eqnal.  (Hee 
Prntt's  Mech.,  p.  438.  For  flirthor  demoi'stratlon  of  this  equality,  tbesindcnt  la 
roftorred  to  Ruuth's  BiKid  Dynamics,  pp.  181'  and  188.) 


r 


n 


=  N.  (1) 


le  same  way. 
moments  and 

0  axes  fixed  in 
e  as  the  body 
velocities  about 

1  space  are  per- 
it  the  principal 
ent  under  con- 
^rt  232), 

=  0; 

meuts  of  inertia 
>me8 


;  and  similarly 

oing  the  letters 
iy,  tOt,  the  three 

OS,  w,  and  <<>«,  darins  a 

of  z,  wiU  dUftr  only  by 
)y  the  axil  of  x,  during 
I*  will  therefore  be  an 
veg  will  bo  eqnal.  (See 
jquallty,  the  (indent  ta 


SULES'S  SqUATTONS. 


499 


equations  of  motion  of  the  body  referred  to  the  principal 
axes  at  the  fixed  point  are 


dt 


/?!!■;;»_  (C  -  4)  UjO),  =2  M, 


(2) 


These  are  called  Eulor's  Equations. 

SoH. — ^If  the  body  is  moving  so  there  is  no  point  in  it 
which  is  fixed  in  space,  the  motion  of  the  body  about  its 
centre  of  gravity  is  the  same  as  if  that  point  were  fixed. 

It  is  clear  that,  instead  of  referring  the  motion  of  the 
body  to  the  principal  axes  at  the  fixed  point,  as  Enler  has 
done,  we  may  use  any  axes  fixed  in  the  body.  But  these 
are  in  general  so  complicated  as  to  be  nearly  useless. 

258.  Motion  of  a  Body  about  a  Principal  Asia 
through  its  Centre  of  Gra^ty. — If  a  body  rotate  about 
one  of  its  principal  axes  pamng  through  the  centre  of 
gravity,  this  axis  mil  suffer  no  pressure  from  the  centrifu- 
gal force. 

Let  the  body  rotate  about  the  axis  of  « ;  then  if  u  be  its 
angular  velocity,  the  centrifugal  force  of  any  particle  m 
will  be  (Art.  198,  Cor.  1) 

nw^p  —  WW*  V'*'  +  y', 

which  gives  for  the  x-  and  y-components  muh;  and  tmchf ; 
and  the  moments  of  these  forces  with  respect  to  the  axes  of 
y  and  x  are  for  the  whole  body 

S,tnuh;x,    and    £m<>>>y«. 


600 


AXIS  OF  PERMANENT  ROTATION. 


But  these  are  each  equal  to  zero  when  the  axis  of  rotation 
is  a  principal  axis  (Art.  a32) ;  hence,  the  centrifugal  force 
will  have  no  tendency  to  incline  the  axis  of  z  towai-ds  the 
plane  of  xy.  In  this  case  the  only  effect  of  the  forces  mtJ^x 
and  mul^y  on  the  axis  U  to  move  it  parallel  to  itself,  or  to 
translate  the  body  in  the  directions  of  x  and  y.  But  the 
sum  of  all  these  forces  is 

^Lmurhi    and    I.miJ'y, 

each  of  which  is  equal  to  zero  when  the  axis  of  rotation 
passes  through  the  centre  of  gravity ;  hence  we  conclude 
that,  wfien  a  body  rotates  about  one  of  its  principal  axes 
passing  through  its  centre  of  gravity,  the  rotation  causes  no 
pressure  upon  the  axis. 

If  the  body  rotates  about  this  axis  it  will  continue  to 
rotate  about  it  if  the  axis  be  removed.  On,  this  account  a 
principal  axis  through  the  centre  of  gravity  is  called  an 
axis  of  permanent  rotation.* 

SoH. — If  the  body  be  free,  and  it  begins  to  rotate  about 
an  axis  very  near  to  a  principal  axis,  the  centrifugal  force 
will  cause  the  axis  of  rotation  to  change  continuidly,  inas- 
much as  the  foregoing  conditions  cannot  obtain,  and  this 
axis  of  rotation  will  either  continually  oscillate  about  the 
principal  axis,  always  remaining  very  near  to  it,  or  else  it 
will  remove  itself  indefinitely  from  the  principal  axis. 
Hence,  whenever  we  observe  a  fVee  body  rotating  about  an 
axis  during  any  time,  however  short,  we  may  infer  that  it 
has  continued  to  rotate  about  that  axis  from  the  beginning 
of  the  motion,  and  that  it  will  continue  to  rotate  about  it 
for  ever,  unless  chocked  by  some  extrajieous  obstacle.  (See 
Young's  Mechs.,  p.  230,  also  Venturoli,  pp.  135  and  160.) 


♦  Prstt's  Mochs.,  p.  489.    Called  also  a  natural  axU  qf  rotatUm,  seo  Tonng's 
Hectw.,  p.  sao ;  aim  (?n  intartailt  aicU,  we  Price'«  Mechs.,  Vol.  n,  p.  997. 


m. 

ixis  of  rotation 
ntrifugal  force 
if  z  towards  the 
the  forces  m(>)^x 
I  to  itself,  or  to 
id  y.    But  the 


ixis  of  rotation 
ce  we  conclude 
( principal  axes 
lation  causes  no 

will  continue  to 

this  account  a 

ity  is  called  an 


to  rotate  about 
entrifugal  force 
ontinuaUy,  inas- 
obtain,  and  this 
cillate  about  the 

to  it,  or  else  it 
)  principal  axis. 
>tating  about  an 
lay  infer  that  it 
m  the  beginning 
»  rotate  about  it 
18  obstacle.  (See 
..  136  and  160.) 


r  rotation,  «eo  Tonng'B 
Vol.  n,  p.  m- 


VBLOCmr  ABOUT  A  PRINCIPAL  AXIS. 


501 


259.  Velocity  about  a  Principal  Axis  when  there 
are  no  Accelerating  Forces. — In  this  case  L  —  M  = 
JV  =  0  in  (2)  of  Art.  257 ;  also  A,  B,  0  axQ  constant  for 
the  same  body ;  and  if  we  put 


B-o^^^  £^  =  0,   ^ 


G 


=  H, 


(1) 


A      -"        B 

(2)  of  Art.  257  becomes 

dui^  =  Htii^u^dt. 

Put  u^fti^u^dt  =  dip,  and  we  have 

a^dt'ix  =  Fdtp.,    <>),fifu>g  =  Od^,    (^idu^  =  ffdip; 

and  integrating,^  we  get 

w,»  =  2i?V>  +  o',  w,»  ==  2(70  +  **,  w,«  =  2Zr0  +  c«.  (2) 

where  a,  &,  c  are  the  initiil  yalues  of  w„  w,,  w,;  hence 
from  (1)  and  (2) 

dt  = 


d^ 


V'(2/l^  +  a»)  (2(70  +  IP)  {2H<p  +  c») 


(3) 


Suppose  now  the  Irody  begins  to  tuni  about  only  one  of 
the  principal  axes,  say  the  axis  of  x,  with  the  angular 
velocity  a,  then  *  =  0,  c  =  0,  and  (3)  becomes 


<?<  = 


d<t> 


2  'S/^OS  0  V^Fp  +  a» 


Replacing  2i^  +  a'  by  its  value  W|«,  and  rf0  by  its  value 


F 


i-,  we  have 


<«  = 


dt^^^ 


Voir  Wi' 


i8' 


602 


THB  INTEORAL  OF  EULER'a  XQUATIONS, 


and  integrating,  we  get 


ti.  —a 


0+ tVGff=~  log^^~. 


w. 


e*i(7 .  giat  ^oa  —  rii 


w,  +o 


(4) 


The  constant  (7  mnst  be  determined  so  that  when  ^  =  0, 
Wj  is  the  initial  velocity  a ;  hence  ^^  =  0  or  (7  =  —  oo , 
which  makes  the  first  member  of  (4)  zero  for  every  value 
of  L  Hence,  at  any  time  t,  we  mnst  have  u^  =  a;  and 
therefore  from  (2)  ^  =  0,  and  w,  =  u,  =  0.  Conse- 
qumily  tits  impressed  velocity  about  one  of  the  principal 
axes  of  rotation  continues  perpetttal  and  uniform,  as  before 
shown  (Art  258). 

260.  The  Integral  of  Euler's  Equations.—^  body 
revolves  about  its  centre  of  gravity  acted  on  by  no  forces  but 
such  as  pass  through  that  point;  to  integrate  the  equations 
of  motion. 

As  the  only  forces  acting  on  the  body  ai'e  those  which 
pass  through  its  centre  of  gravity,  they  create  no  moment 
of  rotation  about  an  axis  passing  through  that  centre;  and 
therefore  (2)  of  Art.  257  become 


A^-(B-C)u>,o>,=0/ 
B^-iC-A)o>,o>,=0,] 


(1) 


the  principal  axes  being  drawn  through  the  centre  of 
gravity. 


Tiosa. 


(4) 

at  when  ^  =  0, 
or  C  =  —  oo , 
for  every  value 
i  u^  =  a',  and 
=  0.  Conse- 
f  the  principal 
iform,  as  before 


dona— ^  body 
>y  no  forces  but 
le  the  equations 

!u'e  those  which 
ate  no  moment 
bat  centre ;  and 


(1) 


the  centre  of 


SraW  INTEQBAL  OF  EULSR'S  EQUATIONS. 


503 


Multiply  these  equations  severally  (1)  by  Wj,  w,,  Wj ; 
and  (2)  by  Au)^,  -Bw,,  Cwg,  and  add ;  then  we  have 


di»>t 


(?6), 


du>t 
dt 


=  0; 


^  +  (^.^  =  0,1 


(2) 


integrating,  we  have 


where  A«  and  **  are  the  constants  of  integration. 
Eliminating  Wj'  from  (3),  we  have 

A{A-C)i^*  +  B{B-  C) 0),'  =  *» -  a»; 


(8) 


I 


"•'  ~  £  (5  -  C) 


[ifc9_oa»_^(^_C)V];  (*) 


anda,,»  =  ^^^^[i»-5A»-^M-P)«,«].     (5) 

Substituting  these  values  of  u,  and  Wj  in  the  first  of  equa- 
tions (1),  we  have 


dt   ^L 


dt 


(A-O)iA-B) 


EG 


r-"-A{A-C)) 


G^ISj-')]*-  <«) 


which  is  generally  an  elliptic  transcendent,  and  so  does  not 
admit  of  integration  in  finite  terms.  In  certain  particular 
cases  it  may  be  integrated,  which  will  give  the  value  of  <«>, 
in  terms  of  /,  and  if  this  value  bo  substituted  in  (4)  and  (5), 


604      APPLICATION  OF  TBS  OSNERAL  EqUATIONS, 

the  values  of  w,  and  w,  in  tonus  of  I  will  be  known,  and 
thus,  in  these  caaes,  the  problem  admits  of  compiute  solu- 
tion. 

Cob. — I^t  Wa,  jwy,  0)8  bo  the  axial  components  of  the 
initial  angular  velocity  about  the  principal  axes  when 
<  =  0;  then  integrating  the  first  of  (2),  and  taking  tiie 
limits  corresponding  to  /  and  0,  we  have 

^w*»  -I-  5a»,a  +  Cw3»  =  Jw,2  +  5a),»  +  Cio?.       (7) 

Ijet  a,  /J,  y  be  the  direction-angles  of  the  instantaneous 
axis  at  the  time  t  relative  to  the  principal  axes ;  so  that,  if 
6)  is  the  instantaneous  angular  velocity,  and  I.mr^  is  the 
moment  of  inertia  relative  to  that  axis,  we  have  (Art.  253), 
b),  =  u  cos  a,  6),  =  (i)  cos  /3,  0),  =  (i>  cos  y,  which  sub- 

-tuted  in  (7),  gives 

J  +  B(^y*  +  Cw,"  =  6)«  ( J  cos^  « +  5  cos"  /3  +  C  COS*  y) 
'     '    ?  =  <JiZmr»  (Art.  332,  Cor.) 


w  ;(>l'^  ■'.:■ 


=  the  vis  viva  of  the  body  ; 

from  which  it  appears  that  the  vis  viva  of  the  body  is  con- 
stant throughout  the  whole  motion.  , 

Rem.— An  application  of  the  general  equations  of  rotatory 
motion  (Art.  267),  which  is  of  great  interest  and  impor- 
tance, is  that  of  the  rotatory  phenomena  of  the  earth  under 
the  action  of  the  attracting  forces  of  the  sun  and  the  moon, 
tlie  rotation  being  considered  relative  to  the  centre  of 
gravity  and  an  axis  passing  through  it,  just  as  if  the  centre 
ef  gravity  was  a  fixed  point  (Art.  249,  Sch.) ;  and  the 
problem  treated  as  purely  a  mathematical  one.  Also,  in 
addition  to  the  sun  and  the  moon,  the  problem  may  be 


qVATIONS. 

be  known,  and 
)f  complete  solu- 


nponenta  of  the 
jipal  axes  when 
,  and  taking  tl)c 


•>?  +  Ci^.\      (7) 

the  instantaneous 
axes ;  so  that,  if 

and  'Lmr^  is  the 
have  (Art.  253), 

los  y,  which  eub- 


coB'/S  +  Cco^y) 
,  Cor.) 

e  bodj ; 

)/  the  body  is  con-r 

ations  of  rotatory 
crest  and  impor- 
»f  the  earth  nnder 
in  and  the  moon, 
to  the  centre  of 
9t  as  if  the  centre 

Sch.)  ;  and  the 
al  one.     Also,  in 

problem  may  be 


'•■^rssiziiiimitri.iKiv.i.mi/fnj^'f  « ir",)  ^  «^v  • 


I'^^/i-iii ,  ."ii^yfJi 


SXAMPLES. 


605 


extended  so  aa  to  include  the  action  of  all  the  other  bodies 
whose  influence  affects  the  motion  of  the  eaith's  rotation. 
In  fact  the  investigation  of  the  motion  of  a  system  of  bodies 
in  space  might  be  continued  at  great  length  ;  but  such 
investigations  would  be  clearly  beyond  the  limits  proposed 
in  this  treatise.  The  student  who  desires  to  continue  this 
interesting  subject,  is  referred  to  more  extended  works.* 

EXAMPLES. 

1.  A  hollow  spherical  shell  is  filled  with  fluid,  and  rolls 
down  a  rough  inclined  plane ;  determine  its  motion. 

Let  M  and  M'  l/C  the  masses  of  the  shell  and  fluid 
respectively,  h  and  h'  their  radii  of  gyration  respectively 
alHJut  a  diameter,  and  a  and  a'  the  radii  of  the  exterior  and 
interior  surfaces  of  the  shell ;  then  using  the  same  nota- 
tion aa  in  Art.  246,  we  have 


(Jf  +  if ')  S  =  (-^  +  M')  gAna-  F. 


df> 


(1) 


As  the  spherical  shell  rotates  in  its  descent  down  the  plane, 
the  fluid  has  only  motion  of  translation ;  so  that  the  equa- 
tion of  rotation  is 

JfP^  =  F«.  (3) 

Multiplying  (1)  by  a'  and  (2)  by  a,  and  adding,  we  have 

[{M  +  M')  d>  +  JfF]  H  =  (J/  +  M')  a\j  sin  «.  (3) 

If  the  interior  were  solid,  and  rigidly  joined  to  the  shell, 
the  equation  of  motion  would  be 

•  See  Price's  Mech'g,  VoL  II,  Pmtt'a  Xeob's,  Bonth's  Rigid  Dynamics,  La  Place's 
Hgcanique  C61este,  ete. 


asB 


506 


KXAKPLMS. 


[{M+M')  fl«+^*»+if' /fc'»]  g  ^  {M+M')  ct-g  sin  «.  (4) 

Integrating  (3)  and  (4)  twice,  and  denoting  by  s  and  «'  the 
BIM1C08  thr  tugli  which  the  centre  moves  during  the  time  / 
in  these  two  cases  respectively,  we  have 


(6) 


80  that  a  greater  space  is  described  by  the  sphere  which  has 
the  fluid  than  by  that  which  has  the  solid  in  its  interior. 

If  the  densities  of  the  solid  and  the  fluid  are  the  same, 
we  have  fiom  (5),  by  Art.  233,  Ex.  14, 

~,  =  y  .  _  q  ,,•    (Price's  Anal  Mechs.,  Vol  II,  p.  368). 

2.  A  homoguneons  spliere  rolls  down  within  a  rongh 
spherical  bowl ;  it  is  ro<{uired  to  determine  the  motion. 

liOt  a  be  the  radius  of  the  sphere,  and  b  the  radius  of  the 
bowl ;  and  let  us  suppose  the  sphere  to  be  placed  in  the 
bowl  at  rest.  Lot  OCQ  =  ^, 
QPA  =  e,  BCO  =  a,  u>  = 
the  angular  velocity  of  the 
ball  about  an  axis  throagh  its 
centre  P,  k  =  the  correspond- 
ing radius  of  gyration ;  0M=: 
X,  MP  =  y  ;  «i  =  the  mass  of 
the  ball.    Then  Fig.  mm 


TO  jTj  =r  —  7?  sin  0  +  /*  cos  0 ; 


♦»  T^  =  /<  008  ^  -f  Fnn  ^  —  mg\ 


(1) 


')(^ga.na.  (4) 

by  s  and  s'  the 
ing  the  time  t 


k^ 


(6)  , 


ihcre  which  has 
its  interior. 
)  are  the  same, 


3l  II,  p.  368). 

itliin  a  rough 
he  motion. 
le  radins  of  the 
I  placed  in  the 


i»; 


mg 


(1) 


BXAMPhES. 


at 


507 
(8) 


Also    X  —   (J  —  a)  sin  ^ ;    if  —  b  —  (b  —  a)  cos  ^ 
...    g=(J-a)co,*g-(S-.)«.*(f);      (4) 

g  =  (»- a)  eiu  **  +  (»-.)  00.  *(*)•.     (6) 


—  CO80  +  ^8in^  =  (i-fl)^. 


m 


(d  —  o)  ^  =  JT—  fw^-  sin  ^ 


(7) 


Now  to  determine  the  angnlar  velocity  of  the  ball,  we  mnst 
estimate  the  angle  described  by  a  fixed  line  in  It,  as  PA, 
from  a  lino  fixed  in  direction,  as  PM,  and  the  ratio  of  the 
infinitesimal  increase  of  this  angle  to  that  of  the  time  will 
be  the  angular  velocity  of  the  ball. 


ti 


dMPA       d4>  .  de 


dt 


~  dt  "^  dt 


Since  the  sphere  does  not  slide,  aO  =  ft  (a  —  ^) ; 


a  —  b  d^ 
a      dt 


dit> a  —  b  d^ 

'**    dt  ~~ir  dp' 


from  (3),  (7),  and  (8)  we  get 


(*-o)^~  -^8in^; 


(•) 


(») 


S08 


BXAMPTjBB. 


(10) 


(11) 


.  • .    {b-a)  (^p  =  — ^  (cos  0  -  COB  «). 

Substituting  (9)  in  (7)  we  have 

F  =  ^mg  Bin  0. 
Substituting  (4),  (9),  (10),  (11)  in  (1)  we  have 

5  =  ^  (17  COB  ^  —  10  COB  «)  J 

therefore  the  pressure  at  the  lowest  point 

=  ^(17-10coH«); 

and  the  pressure  of  the  ball  on  the  bowl  vanishes  when 

cos  ^  =  ^  cos  a. 

Cob. — If  the  ball  rolls  over  u  small  arc  at  the  lowest  part 
of  the  be  ivl,  80  that  «  and  <f>  are  always  small,  cos  «,  and 

cos  </)  may  be  roplaced  by  1  —  —  and  I  —  ?■  respectively  ; 
and  from  (10)  we  have 

(«»  _  ^)i       L^  (*  -  «)J 

•••  *  =  «''««[7-(/i:^)J'' 

thus  the  ball  comes  to  rest  at  points  whoso  angular  distance 
is  rt  on  l>oth  sides  of  0,  the  lowest  point  of  the  bowl ;  and 
the  periodic  time  is 


iS  a). 


(10) 


(11) 


ave 


•)i 


'< 


mishcs  wben 


A  tho  lowest  part 
small,  COB  «,  and 

-  ^  respectively ; 


dt; 


t\ 


)  angular  distance 
of  the  bowl ;  and 


SXAMPLSS. 


509 


therefore  the  oscillations  are  performed  in  the  same  time  aa 
those  of  a  dimple  pendulum  whose  length  is  \  {b  —  a), 
(Art.  194).    (Price's  Anal.  Mech's,  Vol.  II,  p.  369.) 

3.  A  homogeneous  sphere  has  an  angular  velocity  w 
about  its  diameter,  and  gradually  contrs^cts,  remaining 
constantly  homogeneous,  till  it  has  half  the  original 
diameter ;  required  the  final  angular  velocity.     Ans.  4w. 

4.  If  the  earth  were  a  homogeneous  sphere,  at  what  point 
must  it  be  struck,  that  it  may  receive  its  prtsent  viilocity 
of  translation  and  of  rotation,  the  former  being  68000  miles 
per  hour  nearly  ?    Ana.  24  miles  nearly  from  the  centre. 

&.  A  homogeneous  sphere  rolls  down  a  rough  inclined 
plane;  the  inclined  plane  rests  on  a  smooth  horizontal 
plane,  along  which  it  slides  by  reason  of  the  pressure  of  the 
sphere ;  required  the  motions  of  the  inclined  plane  and  of 
the  centre  of  the  sphere. 

Let  m  =  the  mass  of  the  sphere, 
M  =  tho  mass  of  the  inclined 
plane,  a  =  the  radius  of  the  sphere, 
rt  =.  the  angle  of  tho  inclined 
plane,  Q  its  apex ;  0  the  place  of  Q 
when  t  z=z  0;  0'  the  point  on  the 
plaiio  which  was  in  contact  with 
the  point  A  of  the  sphere  when 
/  =  0,  at  which  time  we  may  sup- 
pose all  to  be  at  rest ;  A  CF  =  6,  the  angle  through  which 
tho  sphere  has  revolved  in  the  time  i. 

Let  0  be  the  origin,  and  let  the  horizontal  and  vertical 
lines  through  it  be  tiio  axes  of  x  and  y ;  OQ  =  x' ;  and  -let 
{x,  y)  {h,  k)  be  the  places  of  the  centre  of  tho  sphere  at  the 
times  (  —  t  and  /  =  0  respectively.  Then  the  equations 
of  motion  of  the  sphere  are 

w  j^  =  F  cos  «  —  li  sin  a, 


FI8.I02 


610 


BXAMPLES. 


«» ^  =  ^ein  a  +  /J  COS  o  —  mgt 

and  the  equation  of  motion  of  the  plane  is 

.  M  -^  =  —  F  co^  a  ■{■  R  sax  a. 

From  the  geometry  we  have 

a;  =  A  +  «'  —  aO  cos  o, 

y  =  ifc  —  oO  sin  a. 
From  these  equations  we  obtain 

,       wi  cos  «    - 

__         6ct  sin  a  cos  a  gfi  _ 

~  7  (m  +  My—  bm  cos*^ '  T ' 


«  =  A  — 


y  =  *- 


Slfsinacosa  gfi 

6{m  +  M)  sin*  a         gfi 
7  (m  +  if )  —  5»«  co8»  a  '  aT 


which  give  the  values  of  a;  and  y  in' terms  of  /. 
Also  wo  obtain 

(m  +  M)  {x  —  h)  sin  o  —  if  (y  —  /fc)  cos  a  =  0 ; 

which  is  the  equation  of  the  path  describeti  by  the  centre 
of  tlie  sphere  ;  and  therefore  tliis  path  is  a  straiglit  line. 

6.  A  heavy  solid  wheel  in  tlie  form  of  a  right  circular 
cylinder,  it)  composed  uf  two  substunces,  whuau  volumes  are 


-mg. 


u  <c 


COB  a 


9?., 

bm  cob'  a     "i 


of/. 


fc)  COS  «  =  0 ; 

bed  by  the  centre 
a  straight  line. 

)f  a  right  circular 
whusu  vuluiuea  are 


BXAJtPLES. 


611 


equal,  and  whose  densities  are  p  and  p' ;  these  substances 
are  arranged  in  two  diflferent  forms;  in  one  case,  that  whose 
density  is  p  occupies  the  central  part  of  the  wheel,  and  the 
other  is  placed  as  a  ring  round  it ;  in  the  second  case,  the 
places  of  the  substances  ai-e  interchanged  ;  t  and  t'  are  the 
times  in  which  the  wheels  roll  down  a  given  rou^h  inclined 
plane  from  rest;  show  that 

<»:<'»::  6p  +  7p'  :  5p'  +  7p. 

7.  A  homogeneous  sphere  moves  down  a  rough  inclined 
plane,  whose  angle  of  inclination  «  to  the  horizon  is  greater 
than  that  of  the  angle  of  frictiou ;  it  is  required  to  show  (1) 
that  the  sphere  will  roll  without  sliding  when  ^  is  equal  to 
or  greater  than  f  tan  «,  and  (2)  that  it  will  slide  and  roll 
when  n  is  less  than  f  tan  a,  where  |u  is  the  coefficient  of 
friction. 

b.  In  the  last  example  show  that  the  angular  velocity  of 

.  ,■     .■       A  M^  i.        5/10  cos  a  . 

the  sphere  at  the  time  t  from  rest  =  -•  ■    - —  t. 

9.  If  the  body  moving  down  the  plane  is  a  circular 
cylinder  of  radius  =  a,  with  its  axis  horizontal,  show  that 
the  body  will  slide  and  roll,  or  roll  only,  according  as  a  is 
greater  or  not  greater  than  tan~^  3/i. 


